2isqrt (example 3.6)

Percentage Accurate: 69.9% → 99.7%

Time: 11.2s

Alternatives: 15

Speedup: 1.9×

Specification

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: 69.9% 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: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. add-exp-log 52.0%

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

   2. inv-pow 52.0%

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

   3. pow1/2 52.0%

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

   4. pow-pow 52.0%

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

   5. log-pow 52.0%

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

   6. metadata-eval 52.0%

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

   7. +-commutative 52.0%

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

   8. log1p-udef 52.0%

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

3. Applied egg-rr 52.0%

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

   1. exp-prod 52.3%

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

5. Simplified 52.3%

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

   1. expm1-log1p-u 52.3%

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

   2. expm1-udef 51.9%

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

   3. add-exp-log 51.9%

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

   4. log-pow 51.9%

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

   5. log1p-udef 51.9%

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

   6. add-log-exp 51.9%

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

   7. pow-to-exp 51.9%

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

   8. +-commutative 51.9%

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

7. Applied egg-rr 51.9%

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

   1. expm1-def 63.1%

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

   2. expm1-log1p 63.1%

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

9. Simplified 63.1%

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

    1. flip-- 63.1%

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

    2. frac-times 56.1%

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

    3. metadata-eval 56.1%

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

    4. add-sqr-sqrt 58.2%

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

    5. pow-prod-up 67.7%

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

    6. metadata-eval 67.7%

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

    7. inv-pow 67.7%

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

    8. pow1/2 67.7%

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

    9. pow-flip 67.7%

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

    10. metadata-eval 67.7%

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

    11. un-div-inv 67.6%

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

    12. frac-sub 68.7%

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

    13. *-un-lft-identity 68.7%

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

11. Applied egg-rr 68.8%

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

    1. associate-/r* 68.8%

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

    2. associate-/r/ 68.8%

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

    3. *-commutative 68.8%

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

    4. associate-*r/ 68.8%

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

    5. *-rgt-identity 68.8%

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

    6. associate-/r* 68.8%

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

    7. *-rgt-identity 68.8%

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

    8. associate-/l* 68.8%

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

    9. /-rgt-identity 68.8%

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

    10. *-commutative 68.8%

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

    11. fma-udef 68.8%

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

    12. distribute-lft1-in 68.8%

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

    13. *-commutative 68.8%

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

    14. *-commutative 68.8%

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

    15. associate-*r* 68.8%

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

    16. associate-/l/ 68.8%

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

13. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\frac{1 + x \cdot 0}{x}}{\mathsf{fma}\left({x}^{-0.5}, 1 + x, {\left(1 + x\right)}^{0.5}\right)}} \]
14. Step-by-step derivation

    1. mul0-rgt 99.4%

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

    2. metadata-eval 99.4%

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

    3. inv-pow 99.4%

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

    4. metadata-eval 99.4%

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

    5. pow-sqr 99.6%

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

    6. *-un-lft-identity 99.6%

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

    7. times-frac 99.6%

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

    8. unpow1/2 99.6%

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

15. Applied egg-rr 99.6%

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

16. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. add-exp-log 52.0%

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

   2. inv-pow 52.0%

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

   3. pow1/2 52.0%

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

   4. pow-pow 52.0%

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

   5. log-pow 52.0%

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

   6. metadata-eval 52.0%

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

   7. +-commutative 52.0%

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

   8. log1p-udef 52.0%

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

3. Applied egg-rr 52.0%

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

   1. exp-prod 52.3%

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

5. Simplified 52.3%

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

   1. expm1-log1p-u 52.3%

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

   2. expm1-udef 51.9%

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

   3. add-exp-log 51.9%

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

   4. log-pow 51.9%

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

   5. log1p-udef 51.9%

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

   6. add-log-exp 51.9%

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

   7. pow-to-exp 51.9%

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

   8. +-commutative 51.9%

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

7. Applied egg-rr 51.9%

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

   1. expm1-def 63.1%

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

   2. expm1-log1p 63.1%

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

9. Simplified 63.1%

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

    1. flip-- 63.1%

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

    2. frac-times 56.1%

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

    3. metadata-eval 56.1%

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

    4. add-sqr-sqrt 58.2%

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

    5. pow-prod-up 67.7%

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

    6. metadata-eval 67.7%

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

    7. inv-pow 67.7%

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

    8. pow1/2 67.7%

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

    9. pow-flip 67.7%

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

    10. metadata-eval 67.7%

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

    11. un-div-inv 67.6%

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

    12. frac-sub 68.7%

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

    13. *-un-lft-identity 68.7%

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

11. Applied egg-rr 68.8%

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

    1. associate-/r* 68.8%

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

    2. associate-/r/ 68.8%

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

    3. *-commutative 68.8%

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

    4. associate-*r/ 68.8%

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

    5. *-rgt-identity 68.8%

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

    6. associate-/r* 68.8%

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

    7. *-rgt-identity 68.8%

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

    8. associate-/l* 68.8%

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

    9. /-rgt-identity 68.8%

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

    10. *-commutative 68.8%

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

    11. fma-udef 68.8%

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

    12. distribute-lft1-in 68.8%

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

    13. *-commutative 68.8%

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

    14. *-commutative 68.8%

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

    15. associate-*r* 68.8%

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

    16. associate-/l/ 68.8%

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

13. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\frac{1 + x \cdot 0}{x}}{\mathsf{fma}\left({x}^{-0.5}, 1 + x, {\left(1 + x\right)}^{0.5}\right)}} \]
14. Step-by-step derivation

    1. expm1-log1p-u 96.1%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1 + x \cdot 0}{x}}{\mathsf{fma}\left({x}^{-0.5}, 1 + x, {\left(1 + x\right)}^{0.5}\right)}\right)\right)} \]

    2. expm1-udef 63.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1 + x \cdot 0}{x}}{\mathsf{fma}\left({x}^{-0.5}, 1 + x, {\left(1 + x\right)}^{0.5}\right)}\right)} - 1} \]

    3. clear-num 63.9%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{\mathsf{fma}\left({x}^{-0.5}, 1 + x, {\left(1 + x\right)}^{0.5}\right)}{\frac{1 + x \cdot 0}{x}}}}\right)} - 1 \]

    4. mul0-rgt 63.9%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{1}{\frac{\mathsf{fma}\left({x}^{-0.5}, 1 + x, {\left(1 + x\right)}^{0.5}\right)}{\frac{1 + \color{blue}{0}}{x}}}\right)} - 1 \]

    5. metadata-eval 63.9%

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

    6. div-inv 63.9%

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

    7. unpow1/2 63.9%

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

    8. clear-num 63.9%

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

    9. /-rgt-identity 63.9%

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

15. Applied egg-rr 63.9%

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

    1. expm1-def 95.5%

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

    2. expm1-log1p 98.7%

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

    3. *-commutative 98.7%

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

    4. associate-/r* 99.4%

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

17. Simplified 99.4%

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

18. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1 + x \cdot 0}{x}}{\sqrt{x + 1} + {x}^{-0.5} \cdot \left(x + 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (+ 1.0 (* x 0.0)) x) (+ (sqrt (+ x 1.0)) (* (pow x -0.5) (+ x 1.0)))))

C

double code(double x) {
	return ((1.0 + (x * 0.0)) / x) / (sqrt((x + 1.0)) + (pow(x, -0.5) * (x + 1.0)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 + (x * 0.0d0)) / x) / (sqrt((x + 1.0d0)) + ((x ** (-0.5d0)) * (x + 1.0d0)))
end function

Java

public static double code(double x) {
	return ((1.0 + (x * 0.0)) / x) / (Math.sqrt((x + 1.0)) + (Math.pow(x, -0.5) * (x + 1.0)));
}

Python

def code(x):
	return ((1.0 + (x * 0.0)) / x) / (math.sqrt((x + 1.0)) + (math.pow(x, -0.5) * (x + 1.0)))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 + Float64(x * 0.0)) / x) / Float64(sqrt(Float64(x + 1.0)) + Float64((x ^ -0.5) * Float64(x + 1.0))))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 + (x * 0.0)) / x) / (sqrt((x + 1.0)) + ((x ^ -0.5) * (x + 1.0)));
end

Wolfram

code[x_] := N[(N[(N[(1.0 + N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[(N[Power[x, -0.5], $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.9%

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

   1. add-exp-log 52.0%

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

   2. inv-pow 52.0%

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

   3. pow1/2 52.0%

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

   4. pow-pow 52.0%

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

   5. log-pow 52.0%

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

   6. metadata-eval 52.0%

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

   7. +-commutative 52.0%

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

   8. log1p-udef 52.0%

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

3. Applied egg-rr 52.0%

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

   1. exp-prod 52.3%

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

5. Simplified 52.3%

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

   1. expm1-log1p-u 52.3%

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

   2. expm1-udef 51.9%

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

   3. add-exp-log 51.9%

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

   4. log-pow 51.9%

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

   5. log1p-udef 51.9%

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

   6. add-log-exp 51.9%

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

   7. pow-to-exp 51.9%

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

   8. +-commutative 51.9%

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

7. Applied egg-rr 51.9%

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

   1. expm1-def 63.1%

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

   2. expm1-log1p 63.1%

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

9. Simplified 63.1%

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

    1. flip-- 63.1%

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

    2. frac-times 56.1%

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

    3. metadata-eval 56.1%

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

    4. add-sqr-sqrt 58.2%

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

    5. pow-prod-up 67.7%

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

    6. metadata-eval 67.7%

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

    7. inv-pow 67.7%

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

    8. pow1/2 67.7%

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

    9. pow-flip 67.7%

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

    10. metadata-eval 67.7%

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

    11. un-div-inv 67.6%

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

    12. frac-sub 68.7%

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

    13. *-un-lft-identity 68.7%

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

11. Applied egg-rr 68.8%

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

    1. associate-/r* 68.8%

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

    2. associate-/r/ 68.8%

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

    3. *-commutative 68.8%

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

    4. associate-*r/ 68.8%

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

    5. *-rgt-identity 68.8%

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

    6. associate-/r* 68.8%

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

    7. *-rgt-identity 68.8%

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

    8. associate-/l* 68.8%

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

    9. /-rgt-identity 68.8%

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

    10. *-commutative 68.8%

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

    11. fma-udef 68.8%

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

    12. distribute-lft1-in 68.8%

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

    13. *-commutative 68.8%

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

    14. *-commutative 68.8%

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

    15. associate-*r* 68.8%

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

    16. associate-/l/ 68.8%

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

13. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\frac{1 + x \cdot 0}{x}}{\mathsf{fma}\left({x}^{-0.5}, 1 + x, {\left(1 + x\right)}^{0.5}\right)}} \]
14. Step-by-step derivation

    1. fma-udef 99.4%

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

    2. unpow1/2 99.4%

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

15. Applied egg-rr 99.4%

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

16. Final simplification 99.4%

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

Alternative 4: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;x \leq 80000000:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot x}}{{x}^{-0.5} + \frac{1}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= x 80000000.0)
     (+ (pow x -0.5) (/ -1.0 t_0))
     (/ (/ 1.0 (* x x)) (+ (pow x -0.5) (/ 1.0 t_0))))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (x <= 80000000.0) {
		tmp = pow(x, -0.5) + (-1.0 / t_0);
	} else {
		tmp = (1.0 / (x * x)) / (pow(x, -0.5) + (1.0 / t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (x <= 80000000.0d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / t_0)
    else
        tmp = (1.0d0 / (x * x)) / ((x ** (-0.5d0)) + (1.0d0 / t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (x <= 80000000.0) {
		tmp = Math.pow(x, -0.5) + (-1.0 / t_0);
	} else {
		tmp = (1.0 / (x * x)) / (Math.pow(x, -0.5) + (1.0 / t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if x <= 80000000.0:
		tmp = math.pow(x, -0.5) + (-1.0 / t_0)
	else:
		tmp = (1.0 / (x * x)) / (math.pow(x, -0.5) + (1.0 / t_0))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (x <= 80000000.0)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / t_0));
	else
		tmp = Float64(Float64(1.0 / Float64(x * x)) / Float64((x ^ -0.5) + Float64(1.0 / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (x <= 80000000.0)
		tmp = (x ^ -0.5) + (-1.0 / t_0);
	else
		tmp = (1.0 / (x * x)) / ((x ^ -0.5) + (1.0 / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 80000000.0], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 8e7

   1. Initial program 99.4%

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

      1. add-log-exp 6.4%

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

      2. *-un-lft-identity 6.4%

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

      3. log-prod 6.4%

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

      4. metadata-eval 6.4%

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

      5. add-log-exp 99.4%

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

      6. pow1/2 99.4%

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

      7. pow-flip 99.6%

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

      8. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \frac{1}{\sqrt{x + 1}} \]
   4. Step-by-step derivation

      1. +-lft-identity 99.6%

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

   5. Simplified 99.6%

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

   if 8e7 < x

   1. Initial program 36.8%

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

      1. sub-neg 36.8%

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

      2. flip-+ 36.8%

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

      3. frac-times 18.6%

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

      4. metadata-eval 18.6%

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

      5. add-sqr-sqrt 14.8%

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

      6. distribute-neg-frac 14.8%

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

      7. metadata-eval 14.8%

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

      8. +-commutative 14.8%

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

      9. distribute-neg-frac 14.8%

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

      10. metadata-eval 14.8%

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

      11. +-commutative 14.8%

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

      12. pow1/2 14.8%

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

      13. pow-flip 14.8%

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

      14. metadata-eval 14.8%

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

   3. Applied egg-rr 14.8%

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

      1. associate-*r/ 18.3%

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

      2. associate-*l/ 18.3%

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

      3. metadata-eval 18.3%

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

      4. sub-neg 18.3%

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

      5. distribute-neg-frac 18.3%

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

      6. metadata-eval 18.3%

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

   5. Simplified 18.3%

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

   6. Taylor expanded in x around inf 79.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{{x}^{2}}}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}} \]
   7. Step-by-step derivation

      1. unpow2 79.5%

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

   8. Simplified 79.5%

      \[\leadsto \frac{\color{blue}{\frac{1}{x \cdot x}}}{{x}^{-0.5} + \frac{1}{\sqrt{1 + x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 80000000:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{\sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x \cdot x}}{{x}^{-0.5} + \frac{1}{\sqrt{x + 1}}}\\ \end{array} \]

Alternative 5: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. sub-neg 67.9%

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

   2. flip-+ 67.8%

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

   3. frac-times 58.4%

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

   4. metadata-eval 58.4%

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

   5. add-sqr-sqrt 56.6%

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

   6. distribute-neg-frac 56.6%

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

   7. metadata-eval 56.6%

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

   8. +-commutative 56.6%

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

   9. distribute-neg-frac 56.6%

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

   10. metadata-eval 56.6%

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

   11. +-commutative 56.6%

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

   12. pow1/2 56.6%

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

   13. pow-flip 56.5%

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

   14. metadata-eval 56.5%

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

3. Applied egg-rr 56.5%

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

   1. associate-*r/ 58.3%

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

   2. associate-*l/ 58.3%

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

   3. metadata-eval 58.3%

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

   4. sub-neg 58.3%

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

   5. distribute-neg-frac 58.3%

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

   6. metadata-eval 58.3%

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

5. Simplified 58.3%

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

   1. clear-num 61.0%

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

   2. frac-sub 68.4%

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

   3. *-un-lft-identity 68.4%

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

   4. pow1 68.4%

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

   5. inv-pow 68.4%

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

   6. pow-div 68.1%

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

   7. metadata-eval 68.1%

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

   8. pow2 68.1%

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

   9. add-sqr-sqrt 68.8%

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

   10. +-commutative 68.8%

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

   11. pow1 68.8%

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

   12. inv-pow 68.8%

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

   13. pow-div 68.8%

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

   14. metadata-eval 68.8%

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

   15. pow2 68.8%

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

   16. add-sqr-sqrt 68.8%

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

   17. +-commutative 68.8%

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

7. Applied egg-rr 68.8%

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

8. Taylor expanded in x around 0 89.5%

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

9. Final simplification 89.5%

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

Alternative 6: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + x \cdot 0}{x}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (if (<= x 5.8e+102)
     (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))
     (/ (/ (+ 1.0 (* x 0.0)) x) (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else if (x <= 5.8e+102) {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	} else {
		tmp = ((1.0 + (x * 0.0)) / x) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else if (x <= 5.8d+102) then
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    else
        tmp = ((1.0d0 + (x * 0.0d0)) / x) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else if (x <= 5.8e+102) {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	} else {
		tmp = ((1.0 + (x * 0.0)) / x) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	elif x <= 5.8e+102:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	else:
		tmp = ((1.0 + (x * 0.0)) / x) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	elseif (x <= 5.8e+102)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x * 0.0)) / x) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	elseif (x <= 5.8e+102)
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	else
		tmp = ((1.0 + (x * 0.0)) / x) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[x, 5.8e+102], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.3%

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

   if 1 < x < 5.8000000000000005e102

   1. Initial program 13.2%

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

      1. *-un-lft-identity 13.2%

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

      2. clear-num 13.2%

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

      3. associate-/r/ 13.2%

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

      4. prod-diff 13.2%

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

      5. *-un-lft-identity 13.2%

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

      6. fma-neg 13.2%

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

      7. *-un-lft-identity 13.2%

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

      8. inv-pow 13.2%

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

      9. sqrt-pow2 13.1%

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

      10. metadata-eval 13.1%

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

      11. pow1/2 13.1%

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

      12. pow-flip 13.1%

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

      13. +-commutative 13.1%

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

      14. metadata-eval 13.1%

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

   3. Applied egg-rr 13.1%

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

      1. fma-udef 13.1%

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

      2. distribute-lft1-in 13.1%

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

      3. metadata-eval 13.1%

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

      4. mul0-lft 13.1%

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

      5. +-rgt-identity 13.1%

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

   5. Simplified 13.1%

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

   6. Taylor expanded in x around inf 93.3%

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

   if 5.8000000000000005e102 < x

   1. Initial program 48.7%

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

      1. add-exp-log 4.4%

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

      2. inv-pow 4.4%

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

      3. pow1/2 4.4%

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

      4. pow-pow 4.4%

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

      5. log-pow 4.4%

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

      6. metadata-eval 4.4%

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

      7. +-commutative 4.4%

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

      8. log1p-udef 4.4%

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

   3. Applied egg-rr 4.4%

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

      1. exp-prod 5.4%

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

   5. Simplified 5.4%

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

      1. expm1-log1p-u 5.4%

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

      2. expm1-udef 4.5%

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

      3. add-exp-log 4.5%

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

      4. log-pow 4.5%

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

      5. log1p-udef 4.5%

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

      6. add-log-exp 4.5%

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

      7. pow-to-exp 4.5%

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

      8. +-commutative 4.5%

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

   7. Applied egg-rr 4.5%

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

      1. expm1-def 35.3%

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

      2. expm1-log1p 35.3%

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

   9. Simplified 35.3%

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

       1. flip-- 35.3%

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

       2. frac-times 16.4%

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

       3. metadata-eval 16.4%

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

       4. add-sqr-sqrt 22.2%

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

       5. pow-prod-up 48.7%

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

       6. metadata-eval 48.7%

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

       7. inv-pow 48.7%

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

       8. pow1/2 48.7%

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

       9. pow-flip 48.7%

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

       10. metadata-eval 48.7%

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

       11. un-div-inv 48.7%

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

       12. frac-sub 48.7%

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

       13. *-un-lft-identity 48.7%

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

   11. Applied egg-rr 48.7%

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

       1. associate-/r* 48.7%

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

       2. associate-/r/ 48.7%

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

       3. *-commutative 48.7%

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

       4. associate-*r/ 48.7%

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

       5. *-rgt-identity 48.7%

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

       6. associate-/r* 48.7%

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

       7. *-rgt-identity 48.7%

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

       8. associate-/l* 48.7%

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

       9. /-rgt-identity 48.7%

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

       10. *-commutative 48.7%

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

       11. fma-udef 48.7%

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

       12. distribute-lft1-in 48.7%

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

       13. *-commutative 48.7%

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

       14. *-commutative 48.7%

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

       15. associate-*r* 48.7%

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

       16. associate-/l/ 48.7%

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

   13. Simplified 99.6%

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

   14. Taylor expanded in x around inf 54.9%

       \[\leadsto \frac{\frac{1 + x \cdot 0}{x}}{\color{blue}{\sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + x \cdot 0}{x}}{\sqrt{x}}\\ \end{array} \]

Alternative 7: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 135000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + x \cdot 0}{x}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 135000000.0)
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))
   (if (<= x 5.8e+102)
     (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))
     (/ (/ (+ 1.0 (* x 0.0)) x) (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 135000000.0) {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	} else if (x <= 5.8e+102) {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	} else {
		tmp = ((1.0 + (x * 0.0)) / x) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 135000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    else if (x <= 5.8d+102) then
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    else
        tmp = ((1.0d0 + (x * 0.0d0)) / x) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 135000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	} else if (x <= 5.8e+102) {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	} else {
		tmp = ((1.0 + (x * 0.0)) / x) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 135000000.0:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	elif x <= 5.8e+102:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	else:
		tmp = ((1.0 + (x * 0.0)) / x) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 135000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	elseif (x <= 5.8e+102)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x * 0.0)) / x) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 135000000.0)
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	elseif (x <= 5.8e+102)
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	else
		tmp = ((1.0 + (x * 0.0)) / x) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 135000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e+102], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.35e8

   1. Initial program 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. clear-num 99.4%

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

      3. associate-/r/ 99.4%

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

      4. prod-diff 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. fma-neg 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. inv-pow 99.4%

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

      9. sqrt-pow2 99.6%

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

      10. metadata-eval 99.6%

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

      11. pow1/2 99.6%

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

      12. pow-flip 99.6%

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

      13. +-commutative 99.6%

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

      14. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. fma-udef 99.6%

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

      2. distribute-lft1-in 99.6%

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

      3. metadata-eval 99.6%

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

      4. mul0-lft 99.6%

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

      5. +-rgt-identity 99.6%

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

   5. Simplified 99.6%

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

   if 1.35e8 < x < 5.8000000000000005e102

   1. Initial program 7.2%

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

      1. *-un-lft-identity 7.2%

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

      2. clear-num 7.2%

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

      3. associate-/r/ 7.2%

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

      4. prod-diff 7.2%

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

      5. *-un-lft-identity 7.2%

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

      6. fma-neg 7.2%

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

      7. *-un-lft-identity 7.2%

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

      8. inv-pow 7.2%

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

      9. sqrt-pow2 7.3%

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

      10. metadata-eval 7.3%

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

      11. pow1/2 7.3%

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

      12. pow-flip 7.3%

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

      13. +-commutative 7.3%

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

      14. metadata-eval 7.3%

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

   3. Applied egg-rr 7.3%

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

      1. fma-udef 7.3%

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

      2. distribute-lft1-in 7.3%

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

      3. metadata-eval 7.3%

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

      4. mul0-lft 7.3%

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

      5. +-rgt-identity 7.3%

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

   5. Simplified 7.3%

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

   6. Taylor expanded in x around inf 98.1%

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

   if 5.8000000000000005e102 < x

   1. Initial program 48.7%

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

      1. add-exp-log 4.4%

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

      2. inv-pow 4.4%

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

      3. pow1/2 4.4%

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

      4. pow-pow 4.4%

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

      5. log-pow 4.4%

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

      6. metadata-eval 4.4%

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

      7. +-commutative 4.4%

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

      8. log1p-udef 4.4%

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

   3. Applied egg-rr 4.4%

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

      1. exp-prod 5.4%

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

   5. Simplified 5.4%

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

      1. expm1-log1p-u 5.4%

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

      2. expm1-udef 4.5%

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

      3. add-exp-log 4.5%

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

      4. log-pow 4.5%

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

      5. log1p-udef 4.5%

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

      6. add-log-exp 4.5%

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

      7. pow-to-exp 4.5%

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

      8. +-commutative 4.5%

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

   7. Applied egg-rr 4.5%

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

      1. expm1-def 35.3%

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

      2. expm1-log1p 35.3%

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

   9. Simplified 35.3%

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

       1. flip-- 35.3%

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

       2. frac-times 16.4%

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

       3. metadata-eval 16.4%

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

       4. add-sqr-sqrt 22.2%

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

       5. pow-prod-up 48.7%

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

       6. metadata-eval 48.7%

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

       7. inv-pow 48.7%

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

       8. pow1/2 48.7%

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

       9. pow-flip 48.7%

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

       10. metadata-eval 48.7%

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

       11. un-div-inv 48.7%

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

       12. frac-sub 48.7%

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

       13. *-un-lft-identity 48.7%

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

   11. Applied egg-rr 48.7%

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

       1. associate-/r* 48.7%

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

       2. associate-/r/ 48.7%

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

       3. *-commutative 48.7%

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

       4. associate-*r/ 48.7%

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

       5. *-rgt-identity 48.7%

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

       6. associate-/r* 48.7%

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

       7. *-rgt-identity 48.7%

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

       8. associate-/l* 48.7%

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

       9. /-rgt-identity 48.7%

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

       10. *-commutative 48.7%

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

       11. fma-udef 48.7%

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

       12. distribute-lft1-in 48.7%

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

       13. *-commutative 48.7%

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

       14. *-commutative 48.7%

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

       15. associate-*r* 48.7%

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

       16. associate-/l/ 48.7%

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

   13. Simplified 99.6%

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

   14. Taylor expanded in x around inf 54.9%

       \[\leadsto \frac{\frac{1 + x \cdot 0}{x}}{\color{blue}{\sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 135000000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + x \cdot 0}{x}}{\sqrt{x}}\\ \end{array} \]

Alternative 8: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 48000000:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{\sqrt{x + 1}}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + x \cdot 0}{x}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 48000000.0)
   (+ (pow x -0.5) (/ -1.0 (sqrt (+ x 1.0))))
   (if (<= x 5.8e+102)
     (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))
     (/ (/ (+ 1.0 (* x 0.0)) x) (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 48000000.0) {
		tmp = pow(x, -0.5) + (-1.0 / sqrt((x + 1.0)));
	} else if (x <= 5.8e+102) {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	} else {
		tmp = ((1.0 + (x * 0.0)) / x) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 48000000.0d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / sqrt((x + 1.0d0)))
    else if (x <= 5.8d+102) then
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    else
        tmp = ((1.0d0 + (x * 0.0d0)) / x) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 48000000.0) {
		tmp = Math.pow(x, -0.5) + (-1.0 / Math.sqrt((x + 1.0)));
	} else if (x <= 5.8e+102) {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	} else {
		tmp = ((1.0 + (x * 0.0)) / x) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 48000000.0:
		tmp = math.pow(x, -0.5) + (-1.0 / math.sqrt((x + 1.0)))
	elif x <= 5.8e+102:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	else:
		tmp = ((1.0 + (x * 0.0)) / x) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 48000000.0)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / sqrt(Float64(x + 1.0))));
	elseif (x <= 5.8e+102)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x * 0.0)) / x) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 48000000.0)
		tmp = (x ^ -0.5) + (-1.0 / sqrt((x + 1.0)));
	elseif (x <= 5.8e+102)
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	else
		tmp = ((1.0 + (x * 0.0)) / x) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 48000000.0], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e+102], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 4.8e7

   1. Initial program 99.4%

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

      1. add-log-exp 6.4%

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

      2. *-un-lft-identity 6.4%

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

      3. log-prod 6.4%

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

      4. metadata-eval 6.4%

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

      5. add-log-exp 99.4%

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

      6. pow1/2 99.4%

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

      7. pow-flip 99.6%

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

      8. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \frac{1}{\sqrt{x + 1}} \]
   4. Step-by-step derivation

      1. +-lft-identity 99.6%

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

   5. Simplified 99.6%

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

   if 4.8e7 < x < 5.8000000000000005e102

   1. Initial program 7.2%

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

      1. *-un-lft-identity 7.2%

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

      2. clear-num 7.2%

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

      3. associate-/r/ 7.2%

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

      4. prod-diff 7.2%

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

      5. *-un-lft-identity 7.2%

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

      6. fma-neg 7.2%

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

      7. *-un-lft-identity 7.2%

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

      8. inv-pow 7.2%

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

      9. sqrt-pow2 7.3%

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

      10. metadata-eval 7.3%

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

      11. pow1/2 7.3%

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

      12. pow-flip 7.3%

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

      13. +-commutative 7.3%

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

      14. metadata-eval 7.3%

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

   3. Applied egg-rr 7.3%

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

      1. fma-udef 7.3%

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

      2. distribute-lft1-in 7.3%

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

      3. metadata-eval 7.3%

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

      4. mul0-lft 7.3%

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

      5. +-rgt-identity 7.3%

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

   5. Simplified 7.3%

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

   6. Taylor expanded in x around inf 98.1%

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

   if 5.8000000000000005e102 < x

   1. Initial program 48.7%

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

      1. add-exp-log 4.4%

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

      2. inv-pow 4.4%

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

      3. pow1/2 4.4%

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

      4. pow-pow 4.4%

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

      5. log-pow 4.4%

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

      6. metadata-eval 4.4%

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

      7. +-commutative 4.4%

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

      8. log1p-udef 4.4%

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

   3. Applied egg-rr 4.4%

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

      1. exp-prod 5.4%

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

   5. Simplified 5.4%

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

      1. expm1-log1p-u 5.4%

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

      2. expm1-udef 4.5%

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

      3. add-exp-log 4.5%

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

      4. log-pow 4.5%

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

      5. log1p-udef 4.5%

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

      6. add-log-exp 4.5%

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

      7. pow-to-exp 4.5%

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

      8. +-commutative 4.5%

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

   7. Applied egg-rr 4.5%

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

      1. expm1-def 35.3%

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

      2. expm1-log1p 35.3%

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

   9. Simplified 35.3%

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

       1. flip-- 35.3%

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

       2. frac-times 16.4%

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

       3. metadata-eval 16.4%

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

       4. add-sqr-sqrt 22.2%

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

       5. pow-prod-up 48.7%

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

       6. metadata-eval 48.7%

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

       7. inv-pow 48.7%

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

       8. pow1/2 48.7%

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

       9. pow-flip 48.7%

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

       10. metadata-eval 48.7%

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

       11. un-div-inv 48.7%

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

       12. frac-sub 48.7%

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

       13. *-un-lft-identity 48.7%

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

   11. Applied egg-rr 48.7%

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

       1. associate-/r* 48.7%

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

       2. associate-/r/ 48.7%

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

       3. *-commutative 48.7%

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

       4. associate-*r/ 48.7%

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

       5. *-rgt-identity 48.7%

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

       6. associate-/r* 48.7%

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

       7. *-rgt-identity 48.7%

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

       8. associate-/l* 48.7%

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

       9. /-rgt-identity 48.7%

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

       10. *-commutative 48.7%

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

       11. fma-udef 48.7%

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

       12. distribute-lft1-in 48.7%

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

       13. *-commutative 48.7%

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

       14. *-commutative 48.7%

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

       15. associate-*r* 48.7%

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

       16. associate-/l/ 48.7%

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

   13. Simplified 99.6%

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

   14. Taylor expanded in x around inf 54.9%

       \[\leadsto \frac{\frac{1 + x \cdot 0}{x}}{\color{blue}{\sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 48000000:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{\sqrt{x + 1}}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + x \cdot 0}{x}}{\sqrt{x}}\\ \end{array} \]

Alternative 9: 72.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + x \cdot 0}{x}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ (/ (+ 1.0 (* x 0.0)) x) (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = ((1.0 + (x * 0.0)) / x) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.5d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = ((1.0d0 + (x * 0.0d0)) / x) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.5) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = ((1.0 + (x * 0.0)) / x) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.5:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = ((1.0 + (x * 0.0)) / x) / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x * 0.0)) / x) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.5)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = ((1.0 + (x * 0.0)) / x) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.5], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(1.0 + N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. inv-pow 99.7%

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

      9. sqrt-pow2 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. fma-udef 100.0%

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

      2. distribute-lft1-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. mul0-lft 100.0%

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

      5. +-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.3%

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

   if 1.5 < x

   1. Initial program 37.9%

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

      1. add-exp-log 7.1%

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

      2. inv-pow 7.1%

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

      3. pow1/2 7.1%

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

      4. pow-pow 7.1%

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

      5. log-pow 7.1%

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

      6. metadata-eval 7.1%

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

      7. +-commutative 7.1%

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

      8. log1p-udef 7.1%

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

   3. Applied egg-rr 7.1%

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

      1. exp-prod 7.9%

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

   5. Simplified 7.9%

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

      1. expm1-log1p-u 7.9%

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

      2. expm1-udef 7.0%

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

      3. add-exp-log 7.0%

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

      4. log-pow 7.0%

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

      5. log1p-udef 7.0%

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

      6. add-log-exp 7.0%

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

      7. pow-to-exp 7.0%

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

      8. +-commutative 7.0%

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

   7. Applied egg-rr 7.0%

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

      1. expm1-def 28.7%

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

      2. expm1-log1p 28.7%

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

   9. Simplified 28.7%

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

       1. flip-- 28.7%

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

       2. frac-times 15.6%

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

       3. metadata-eval 15.6%

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

       4. add-sqr-sqrt 19.6%

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

       5. pow-prod-up 38.1%

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

       6. metadata-eval 38.1%

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

       7. inv-pow 38.1%

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

       8. pow1/2 38.1%

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

       9. pow-flip 38.1%

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

       10. metadata-eval 38.1%

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

       11. un-div-inv 38.1%

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

       12. frac-sub 40.2%

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

       13. *-un-lft-identity 40.2%

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

   11. Applied egg-rr 40.2%

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

       1. associate-/r* 40.2%

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

       2. associate-/r/ 40.2%

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

       3. *-commutative 40.2%

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

       4. associate-*r/ 40.2%

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

       5. *-rgt-identity 40.2%

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

       6. associate-/r* 40.2%

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

       7. *-rgt-identity 40.2%

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

       8. associate-/l* 40.2%

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

       9. /-rgt-identity 40.2%

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

       10. *-commutative 40.2%

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

       11. fma-udef 40.2%

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

       12. distribute-lft1-in 40.2%

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

       13. *-commutative 40.2%

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

       14. *-commutative 40.2%

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

       15. associate-*r* 40.2%

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

       16. associate-/l/ 40.2%

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

   13. Simplified 99.5%

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

   14. Taylor expanded in x around inf 43.9%

       \[\leadsto \frac{\frac{1 + x \cdot 0}{x}}{\color{blue}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + x \cdot 0}{x}}{\sqrt{x}}\\ \end{array} \]

Alternative 10: 69.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. *-un-lft-identity 67.9%

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

   2. clear-num 67.9%

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

   3. associate-/r/ 67.9%

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

   4. prod-diff 67.9%

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

   5. *-un-lft-identity 67.9%

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

   6. fma-neg 67.9%

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

   7. *-un-lft-identity 67.9%

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

   8. inv-pow 67.9%

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

   9. sqrt-pow2 63.2%

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

   10. metadata-eval 63.2%

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

   11. pow1/2 63.2%

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

   12. pow-flip 68.0%

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

   13. +-commutative 68.0%

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

   14. metadata-eval 68.0%

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

3. Applied egg-rr 68.0%

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

   1. fma-udef 68.0%

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

   2. distribute-lft1-in 68.0%

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

   3. metadata-eval 68.0%

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

   4. mul0-lft 68.0%

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

   5. +-rgt-identity 68.0%

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

5. Simplified 68.0%

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

7. Applied egg-rr 67.6%

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

8. Taylor expanded in x around 0 66.0%

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

   1. *-commutative 66.0%

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

   2. frac-sub 66.0%

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

   3. frac-times 66.0%

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

   4. *-un-lft-identity 66.0%

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

   5. *-un-lft-identity 66.0%

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

   6. *-rgt-identity 66.0%

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

   7. +-commutative 66.0%

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

   8. +-commutative 66.0%

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

   9. +-commutative 66.0%

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

10. Applied egg-rr 66.0%

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

    1. associate--l+ 66.7%

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

    2. +-inverses 66.7%

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

    3. metadata-eval 66.7%

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

    4. associate-*r* 66.7%

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

    5. associate-/r* 66.7%

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

    6. *-commutative 66.7%

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

    7. distribute-rgt-in 66.7%

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

    8. *-lft-identity 66.7%

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

    9. pow-plus 66.9%

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

    10. metadata-eval 66.9%

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

    11. unpow1/2 66.9%

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

12. Simplified 66.9%

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

13. Final simplification 66.9%

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

Alternative 11: 67.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;-{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154) (sqrt (/ 1.0 x)) (- (pow (* x x) -0.25))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = sqrt((1.0 / x));
	} else {
		tmp = -pow((x * x), -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.35d+154) then
        tmp = sqrt((1.0d0 / x))
    else
        tmp = -((x * x) ** (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = Math.sqrt((1.0 / x));
	} else {
		tmp = -Math.pow((x * x), -0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35e+154:
		tmp = math.sqrt((1.0 / x))
	else:
		tmp = -math.pow((x * x), -0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = sqrt(Float64(1.0 / x));
	else
		tmp = Float64(-(Float64(x * x) ^ -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.35e+154)
		tmp = sqrt((1.0 / x));
	else
		tmp = -((x * x) ^ -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision], (-N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 69.6%

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

      1. add-log-exp 5.4%

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

      2. *-un-lft-identity 5.4%

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

      3. log-prod 5.4%

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

      4. metadata-eval 5.4%

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

      5. add-log-exp 69.6%

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

      6. pow1/2 69.6%

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

      7. pow-flip 69.8%

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

      8. metadata-eval 69.8%

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

   3. Applied egg-rr 69.8%

      \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \frac{1}{\sqrt{x + 1}} \]
   4. Step-by-step derivation

      1. +-lft-identity 69.8%

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

   5. Simplified 69.8%

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

   6. Taylor expanded in x around inf 66.9%

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

   if 1.35000000000000003e154 < x

   1. Initial program 63.1%

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

      1. add-exp-log 4.4%

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

      2. inv-pow 4.4%

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

      3. pow1/2 4.4%

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

      4. pow-pow 4.4%

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

      5. log-pow 4.4%

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

      6. metadata-eval 4.4%

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

      7. +-commutative 4.4%

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

      8. log1p-udef 4.4%

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

   3. Applied egg-rr 4.4%

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

      1. exp-prod 5.8%

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

   5. Simplified 5.8%

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

   6. Taylor expanded in x around inf 4.0%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 4.0%

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

   8. Simplified 4.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{1}{x}}} \]
   9. Step-by-step derivation

      1. inv-pow 4.0%

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

      2. sqrt-pow1 4.0%

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

      3. metadata-eval 4.0%

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

      4. sqr-pow 4.0%

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

      5. pow-prod-down 63.1%

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

      6. metadata-eval 63.1%

         \[\leadsto -{\left(x \cdot x\right)}^{\color{blue}{-0.25}} \]

   10. Applied egg-rr 63.1%

       \[\leadsto -\color{blue}{{\left(x \cdot x\right)}^{-0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;-{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \]

Alternative 12: 68.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;-{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.35e+154) (/ 1.0 (+ x (sqrt x))) (- (pow (* x x) -0.25))))

C

double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (x + sqrt(x));
	} else {
		tmp = -pow((x * x), -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.35d+154) then
        tmp = 1.0d0 / (x + sqrt(x))
    else
        tmp = -((x * x) ** (-0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.35e+154) {
		tmp = 1.0 / (x + Math.sqrt(x));
	} else {
		tmp = -Math.pow((x * x), -0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.35e+154:
		tmp = 1.0 / (x + math.sqrt(x))
	else:
		tmp = -math.pow((x * x), -0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.35e+154)
		tmp = Float64(1.0 / Float64(x + sqrt(x)));
	else
		tmp = Float64(-(Float64(x * x) ^ -0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.35e+154)
		tmp = 1.0 / (x + sqrt(x));
	else
		tmp = -((x * x) ^ -0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.35e+154], N[(1.0 / N[(x + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Power[N[(x * x), $MachinePrecision], -0.25], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 1.35000000000000003e154

   1. Initial program 69.6%

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

      1. *-un-lft-identity 69.6%

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

      2. clear-num 69.6%

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

      3. associate-/r/ 69.6%

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

      4. prod-diff 69.6%

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

      5. *-un-lft-identity 69.6%

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

      6. fma-neg 69.6%

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

      7. *-un-lft-identity 69.6%

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

      8. inv-pow 69.6%

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

      9. sqrt-pow2 69.8%

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

      10. metadata-eval 69.8%

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

      11. pow1/2 69.8%

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

      12. pow-flip 69.8%

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

      13. +-commutative 69.8%

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

      14. metadata-eval 69.8%

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

   3. Applied egg-rr 69.8%

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

      1. fma-udef 69.8%

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

      2. distribute-lft1-in 69.8%

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

      3. metadata-eval 69.8%

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

      4. mul0-lft 69.8%

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

      5. +-rgt-identity 69.8%

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

   5. Simplified 69.8%

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

   7. Applied egg-rr 69.3%

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

   8. Taylor expanded in x around 0 67.0%

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

   9. Taylor expanded in x around 0 67.9%

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

       1. +-commutative 67.9%

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

       2. *-commutative 67.9%

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

       3. distribute-rgt-in 67.9%

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

       4. *-lft-identity 67.9%

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

       5. pow-plus 68.2%

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

       6. metadata-eval 68.2%

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

       7. unpow1/2 68.2%

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

   11. Simplified 68.2%

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

   if 1.35000000000000003e154 < x

   1. Initial program 63.1%

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

      1. add-exp-log 4.4%

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

      2. inv-pow 4.4%

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

      3. pow1/2 4.4%

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

      4. pow-pow 4.4%

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

      5. log-pow 4.4%

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

      6. metadata-eval 4.4%

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

      7. +-commutative 4.4%

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

      8. log1p-udef 4.4%

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

   3. Applied egg-rr 4.4%

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

      1. exp-prod 5.8%

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

   5. Simplified 5.8%

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

   6. Taylor expanded in x around inf 4.0%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 4.0%

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

   8. Simplified 4.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{1}{x}}} \]
   9. Step-by-step derivation

      1. inv-pow 4.0%

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

      2. sqrt-pow1 4.0%

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

      3. metadata-eval 4.0%

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

      4. sqr-pow 4.0%

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

      5. pow-prod-down 63.1%

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

      6. metadata-eval 63.1%

         \[\leadsto -{\left(x \cdot x\right)}^{\color{blue}{-0.25}} \]

   10. Applied egg-rr 63.1%

       \[\leadsto -\color{blue}{{\left(x \cdot x\right)}^{-0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{x + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;-{\left(x \cdot x\right)}^{-0.25}\\ \end{array} \]

Alternative 13: 50.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.9%

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

   1. add-log-exp 5.0%

      \[\leadsto \color{blue}{\log \left(e^{\frac{1}{\sqrt{x}}}\right)} - \frac{1}{\sqrt{x + 1}} \]

   2. *-un-lft-identity 5.0%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{\frac{1}{\sqrt{x}}}\right)} - \frac{1}{\sqrt{x + 1}} \]

   3. log-prod 5.0%

      \[\leadsto \color{blue}{\left(\log 1 + \log \left(e^{\frac{1}{\sqrt{x}}}\right)\right)} - \frac{1}{\sqrt{x + 1}} \]

   4. metadata-eval 5.0%

      \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{\sqrt{x}}}\right)\right) - \frac{1}{\sqrt{x + 1}} \]

   5. add-log-exp 67.9%

      \[\leadsto \left(0 + \color{blue}{\frac{1}{\sqrt{x}}}\right) - \frac{1}{\sqrt{x + 1}} \]

   6. pow1/2 67.9%

      \[\leadsto \left(0 + \frac{1}{\color{blue}{{x}^{0.5}}}\right) - \frac{1}{\sqrt{x + 1}} \]

   7. pow-flip 63.2%

      \[\leadsto \left(0 + \color{blue}{{x}^{\left(-0.5\right)}}\right) - \frac{1}{\sqrt{x + 1}} \]

   8. metadata-eval 63.2%

      \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) - \frac{1}{\sqrt{x + 1}} \]

3. Applied egg-rr 63.2%

   \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} - \frac{1}{\sqrt{x + 1}} \]
4. Step-by-step derivation

   1. +-lft-identity 63.2%

      \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

5. Simplified 63.2%

   \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{\sqrt{x + 1}} \]

6. Taylor expanded in x around inf 50.0%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{x}}} \]

7. Final simplification 50.0%

   \[\leadsto \sqrt{\frac{1}{x}} \]

Alternative 14: 3.9% accurate, 69.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.5))

C

double code(double x) {
	return x * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.5d0
end function

Java

public static double code(double x) {
	return x * 0.5;
}

Python

def code(x):
	return x * 0.5

Julia

function code(x)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.5;
end

Wolfram

code[x_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 67.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
2. Step-by-step derivation

   1. *-un-lft-identity 67.9%

      \[\leadsto \color{blue}{1 \cdot \frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}} \]

   2. clear-num 67.9%

      \[\leadsto 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\frac{\sqrt{x + 1}}{1}}} \]

   3. associate-/r/ 67.9%

      \[\leadsto 1 \cdot \frac{1}{\sqrt{x}} - \color{blue}{\frac{1}{\sqrt{x + 1}} \cdot 1} \]

   4. prod-diff 67.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -1 \cdot \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right)} \]

   5. *-un-lft-identity 67.9%

      \[\leadsto \mathsf{fma}\left(1, \frac{1}{\sqrt{x}}, -\color{blue}{\frac{1}{\sqrt{x + 1}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   6. fma-neg 67.9%

      \[\leadsto \color{blue}{\left(1 \cdot \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}}\right)} + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   7. *-un-lft-identity 67.9%

      \[\leadsto \left(\color{blue}{\frac{1}{\sqrt{x}}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   8. inv-pow 67.9%

      \[\leadsto \left(\color{blue}{{\left(\sqrt{x}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   9. sqrt-pow2 63.2%

      \[\leadsto \left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   10. metadata-eval 63.2%

       \[\leadsto \left({x}^{\color{blue}{-0.5}} - \frac{1}{\sqrt{x + 1}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   11. pow1/2 63.2%

       \[\leadsto \left({x}^{-0.5} - \frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   12. pow-flip 68.0%

       \[\leadsto \left({x}^{-0.5} - \color{blue}{{\left(x + 1\right)}^{\left(-0.5\right)}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   13. +-commutative 68.0%

       \[\leadsto \left({x}^{-0.5} - {\color{blue}{\left(1 + x\right)}}^{\left(-0.5\right)}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

   14. metadata-eval 68.0%

       \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{\color{blue}{-0.5}}\right) + \mathsf{fma}\left(-1, \frac{1}{\sqrt{x + 1}}, 1 \cdot \frac{1}{\sqrt{x + 1}}\right) \]

3. Applied egg-rr 68.0%

   \[\leadsto \color{blue}{\left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \mathsf{fma}\left(-1, {\left(1 + x\right)}^{-0.5}, {\left(1 + x\right)}^{-0.5}\right)} \]
4. Step-by-step derivation

   1. fma-udef 68.0%

      \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left(-1 \cdot {\left(1 + x\right)}^{-0.5} + {\left(1 + x\right)}^{-0.5}\right)} \]

   2. distribute-lft1-in 68.0%

      \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{\left(-1 + 1\right) \cdot {\left(1 + x\right)}^{-0.5}} \]

   3. metadata-eval 68.0%

      \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} \]

   4. mul0-lft 68.0%

      \[\leadsto \left({x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\right) + \color{blue}{0} \]

   5. +-rgt-identity 68.0%

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

5. Simplified 68.0%

   \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]

6. Taylor expanded in x around 0 49.8%

   \[\leadsto \color{blue}{\left(0.5 \cdot x + {x}^{-0.5}\right) - 1} \]

7. Taylor expanded in x around inf 3.8%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
8. Step-by-step derivation

   1. *-commutative 3.8%

      \[\leadsto \color{blue}{x \cdot 0.5} \]

9. Simplified 3.8%

   \[\leadsto \color{blue}{x \cdot 0.5} \]

10. Final simplification 3.8%

    \[\leadsto x \cdot 0.5 \]

Alternative 15: 1.9% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

double code(double x) {
	return -1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -1.0d0
end function

Java

public static double code(double x) {
	return -1.0;
}

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

function tmp = code(x)
	tmp = -1.0;
end

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 67.9%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

2. Taylor expanded in x around 0 49.2%

   \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{1} \]

3. Taylor expanded in x around inf 2.0%

   \[\leadsto \color{blue}{-1} \]

4. Final simplification 2.0%

   \[\leadsto -1 \]

Developer target: 99.1% 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 2023268 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

  (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ x 1.0)))))