2isqrt (example 3.6)

Percentage Accurate: 68.9% → 94.7%

Time: 11.3s

Alternatives: 11

Speedup: 2.0×

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: 68.9% 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: 94.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right)\\ \frac{1}{t_0} \cdot \frac{\frac{1}{x}}{t_0 \cdot \left(1 + x\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (hypot (pow (hypot 1.0 (sqrt x)) -0.5) (pow x -0.25))))
   (* (/ 1.0 t_0) (/ (/ 1.0 x) (* t_0 (+ 1.0 x))))))

C

double code(double x) {
	double t_0 = hypot(pow(hypot(1.0, sqrt(x)), -0.5), pow(x, -0.25));
	return (1.0 / t_0) * ((1.0 / x) / (t_0 * (1.0 + x)));
}

Java

public static double code(double x) {
	double t_0 = Math.hypot(Math.pow(Math.hypot(1.0, Math.sqrt(x)), -0.5), Math.pow(x, -0.25));
	return (1.0 / t_0) * ((1.0 / x) / (t_0 * (1.0 + x)));
}

Python

def code(x):
	t_0 = math.hypot(math.pow(math.hypot(1.0, math.sqrt(x)), -0.5), math.pow(x, -0.25))
	return (1.0 / t_0) * ((1.0 / x) / (t_0 * (1.0 + x)))

Julia

function code(x)
	t_0 = hypot((hypot(1.0, sqrt(x)) ^ -0.5), (x ^ -0.25))
	return Float64(Float64(1.0 / t_0) * Float64(Float64(1.0 / x) / Float64(t_0 * Float64(1.0 + x))))
end

MATLAB

function tmp = code(x)
	t_0 = hypot((hypot(1.0, sqrt(x)) ^ -0.5), (x ^ -0.25));
	tmp = (1.0 / t_0) * ((1.0 / x) / (t_0 * (1.0 + x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Power[N[Sqrt[1.0 ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision], -0.5], $MachinePrecision] ^ 2 + N[Power[x, -0.25], $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] / N[(t$95$0 * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 67.6%

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

   1. sub-neg 67.6%

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

   2. flip-+ 67.6%

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

   3. frac-times 63.1%

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

   4. metadata-eval 63.1%

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

   5. add-sqr-sqrt 60.1%

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

   6. distribute-neg-frac 60.1%

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

   7. metadata-eval 60.1%

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

   8. +-commutative 60.1%

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

   9. distribute-neg-frac 60.1%

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

   10. metadata-eval 60.1%

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

   11. +-commutative 60.1%

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

   12. pow1/2 60.1%

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

   13. pow-flip 60.1%

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

   14. metadata-eval 60.1%

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

3. Applied egg-rr 60.1%

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

   1. associate-*r/ 61.1%

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

   2. associate-*l/ 61.1%

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

   3. metadata-eval 61.1%

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

   4. associate-/l/ 61.2%

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

   5. rem-square-sqrt 67.5%

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

   6. sub-neg 67.5%

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

   7. distribute-neg-frac 67.5%

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

   8. metadata-eval 67.5%

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

   9. sub-neg 67.5%

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

   10. distribute-neg-frac 67.5%

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

   11. metadata-eval 67.5%

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

5. Simplified 67.5%

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

   1. frac-add 68.5%

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

   2. div-inv 68.5%

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

   3. *-un-lft-identity 68.5%

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

   4. associate-+l+ 89.6%

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

   5. metadata-eval 89.6%

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

   6. frac-times 90.4%

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

   7. un-div-inv 90.4%

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

7. Applied egg-rr 90.4%

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

   1. *-un-lft-identity 90.4%

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

   2. add-sqr-sqrt 90.0%

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

   3. times-frac 90.1%

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

9. Applied egg-rr 89.4%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right)} \cdot \frac{\frac{1 + x \cdot 0}{\left(1 + x\right) \cdot x}}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right)}} \]
10. Step-by-step derivation

    1. mul0-rgt 89.4%

       \[\leadsto \frac{1}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right)} \cdot \frac{\frac{1 + \color{blue}{0}}{\left(1 + x\right) \cdot x}}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right)} \]

    2. metadata-eval 89.4%

       \[\leadsto \frac{1}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right)} \cdot \frac{\frac{\color{blue}{1}}{\left(1 + x\right) \cdot x}}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right)} \]

    3. associate-/l/ 90.2%

       \[\leadsto \frac{1}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right)} \cdot \frac{\color{blue}{\frac{\frac{1}{x}}{1 + x}}}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right)} \]

    4. associate-/l/ 92.3%

       \[\leadsto \frac{1}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right)} \cdot \color{blue}{\frac{\frac{1}{x}}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right) \cdot \left(1 + x\right)}} \]

11. Simplified 92.3%

    \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right)} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right) \cdot \left(1 + x\right)}} \]

12. Final simplification 92.3%

    \[\leadsto \frac{1}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right)} \cdot \frac{\frac{1}{x}}{\mathsf{hypot}\left({\left(\mathsf{hypot}\left(1, \sqrt{x}\right)\right)}^{-0.5}, {x}^{-0.25}\right) \cdot \left(1 + x\right)} \]

Alternative 2: 91.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. sub-neg 67.6%

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

   2. flip-+ 67.6%

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

   3. frac-times 63.1%

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

   4. metadata-eval 63.1%

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

   5. add-sqr-sqrt 60.1%

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

   6. distribute-neg-frac 60.1%

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

   7. metadata-eval 60.1%

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

   8. +-commutative 60.1%

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

   9. distribute-neg-frac 60.1%

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

   10. metadata-eval 60.1%

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

   11. +-commutative 60.1%

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

   12. pow1/2 60.1%

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

   13. pow-flip 60.1%

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

   14. metadata-eval 60.1%

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

3. Applied egg-rr 60.1%

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

   1. associate-*r/ 61.1%

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

   2. associate-*l/ 61.1%

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

   3. metadata-eval 61.1%

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

   4. associate-/l/ 61.2%

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

   5. rem-square-sqrt 67.5%

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

   6. sub-neg 67.5%

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

   7. distribute-neg-frac 67.5%

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

   8. metadata-eval 67.5%

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

   9. sub-neg 67.5%

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

   10. distribute-neg-frac 67.5%

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

   11. metadata-eval 67.5%

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

5. Simplified 67.5%

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

   1. frac-add 68.5%

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

   2. div-inv 68.5%

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

   3. *-un-lft-identity 68.5%

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

   4. associate-+l+ 89.6%

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

   5. metadata-eval 89.6%

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

   6. frac-times 90.4%

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

   7. un-div-inv 90.4%

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

7. Applied egg-rr 90.4%

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

8. Final simplification 90.4%

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

Alternative 3: 91.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. sub-neg 67.6%

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

   2. flip-+ 67.6%

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

   3. frac-times 63.1%

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

   4. metadata-eval 63.1%

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

   5. add-sqr-sqrt 60.1%

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

   6. distribute-neg-frac 60.1%

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

   7. metadata-eval 60.1%

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

   8. +-commutative 60.1%

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

   9. distribute-neg-frac 60.1%

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

   10. metadata-eval 60.1%

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

   11. +-commutative 60.1%

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

   12. pow1/2 60.1%

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

   13. pow-flip 60.1%

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

   14. metadata-eval 60.1%

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

3. Applied egg-rr 60.1%

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

   1. associate-*r/ 61.1%

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

   2. associate-*l/ 61.1%

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

   3. metadata-eval 61.1%

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

   4. associate-/l/ 61.2%

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

   5. rem-square-sqrt 67.5%

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

   6. sub-neg 67.5%

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

   7. distribute-neg-frac 67.5%

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

   8. metadata-eval 67.5%

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

   9. sub-neg 67.5%

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

   10. distribute-neg-frac 67.5%

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

   11. metadata-eval 67.5%

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

5. Simplified 67.5%

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

   1. frac-add 68.5%

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

   2. div-inv 68.5%

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

   3. *-un-lft-identity 68.5%

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

   4. associate-+l+ 89.6%

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

   5. metadata-eval 89.6%

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

   6. frac-times 90.4%

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

   7. un-div-inv 90.4%

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

7. Applied egg-rr 90.4%

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

   1. expm1-log1p-u 87.1%

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

   2. expm1-udef 63.7%

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

9. Applied egg-rr 63.7%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1 + x \cdot 0}{\left(1 + x\right) \cdot x}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 86.3%

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

    2. expm1-log1p 89.6%

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

    3. mul0-rgt 89.6%

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

    4. metadata-eval 89.6%

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

    5. associate-/l/ 90.4%

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

11. Simplified 90.4%

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

12. Final simplification 90.4%

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

Alternative 4: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.0)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (if (<= x 5.5e+102)
     (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))
     (/ 1.0 (pow x 1.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else if (x <= 5.5e+102) {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	} else {
		tmp = 1.0 / pow(x, 1.5);
	}
	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 <= 5.5d+102) then
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    else
        tmp = 1.0d0 / (x ** 1.5d0)
    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 <= 5.5e+102) {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	} else {
		tmp = 1.0 / Math.pow(x, 1.5);
	}
	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 <= 5.5e+102:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	else:
		tmp = 1.0 / math.pow(x, 1.5)
	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 <= 5.5e+102)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	else
		tmp = Float64(1.0 / (x ^ 1.5));
	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 <= 5.5e+102)
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	else
		tmp = 1.0 / (x ^ 1.5);
	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, 5.5e+102], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 100.0%

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

      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. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. mul0-lft 100.0%

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

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.8%

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

   if 1 < x < 5.49999999999999981e102

   1. Initial program 13.7%

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

      1. *-un-lft-identity 13.7%

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

      2. clear-num 13.7%

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

      3. associate-/r/ 13.7%

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

      4. prod-diff 13.7%

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

      5. *-un-lft-identity 13.7%

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

      6. fma-neg 13.7%

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

      7. *-un-lft-identity 13.7%

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

      8. pow1/2 13.7%

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

      9. pow-flip 13.9%

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

      10. metadata-eval 13.9%

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

      11. pow1/2 13.9%

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

      12. pow-flip 14.1%

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

      13. +-commutative 14.1%

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

      14. metadata-eval 14.1%

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

   3. Applied egg-rr 14.1%

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

      1. associate-+l- 14.1%

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

      2. expm1-log1p 14.1%

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

      3. expm1-def 13.3%

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

      4. associate--l- 13.3%

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

      5. fma-udef 13.3%

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

      6. distribute-lft1-in 13.3%

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

      7. metadata-eval 13.3%

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

      8. mul0-lft 13.3%

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

      9. metadata-eval 13.3%

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

      10. expm1-def 14.1%

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

      11. expm1-log1p 14.1%

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

   5. Simplified 14.1%

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

   6. Taylor expanded in x around inf 92.3%

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

   if 5.49999999999999981e102 < x

   1. Initial program 47.1%

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

      1. sub-neg 47.1%

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

      2. flip-+ 47.1%

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

      3. frac-times 33.7%

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

      4. metadata-eval 33.7%

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

      5. add-sqr-sqrt 25.0%

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

      6. distribute-neg-frac 25.0%

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

      7. metadata-eval 25.0%

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

      8. +-commutative 25.0%

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

      9. distribute-neg-frac 25.0%

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

      10. metadata-eval 25.0%

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

      11. +-commutative 25.0%

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

      12. pow1/2 25.0%

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

      13. pow-flip 25.0%

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

      14. metadata-eval 25.0%

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

   3. Applied egg-rr 25.0%

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

      1. associate-*r/ 28.0%

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

      2. associate-*l/ 28.0%

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

      3. metadata-eval 28.0%

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

      4. associate-/l/ 28.5%

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

      5. rem-square-sqrt 47.1%

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

      6. sub-neg 47.1%

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

      7. distribute-neg-frac 47.1%

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

      8. metadata-eval 47.1%

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

      9. sub-neg 47.1%

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

      10. distribute-neg-frac 47.1%

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

      11. metadata-eval 47.1%

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

   5. Simplified 47.1%

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

   6. Taylor expanded in x around inf 47.1%

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

      1. sqrt-div 47.1%

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

      2. metadata-eval 47.1%

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

      3. sqrt-pow1 53.2%

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

      4. metadata-eval 53.2%

         \[\leadsto \frac{1}{{x}^{\color{blue}{1.5}}} \]

   8. Applied egg-rr 53.2%

      \[\leadsto \color{blue}{\frac{1}{{x}^{1.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{1.5}}\\ \end{array} \]

Alternative 5: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 142000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 142000000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (if (<= x 5.5e+102)
     (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))
     (/ 1.0 (pow x 1.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 142000000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else if (x <= 5.5e+102) {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	} else {
		tmp = 1.0 / pow(x, 1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 142000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else if (x <= 5.5d+102) then
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    else
        tmp = 1.0d0 / (x ** 1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 142000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else if (x <= 5.5e+102) {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	} else {
		tmp = 1.0 / Math.pow(x, 1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 142000000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	elif x <= 5.5e+102:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	else:
		tmp = 1.0 / math.pow(x, 1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 142000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	elseif (x <= 5.5e+102)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	else
		tmp = Float64(1.0 / (x ^ 1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 142000000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	elseif (x <= 5.5e+102)
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	else
		tmp = 1.0 / (x ^ 1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 142000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.5e+102], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.42e8

   1. Initial program 98.6%

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

      1. *-un-lft-identity 98.6%

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

      2. clear-num 98.6%

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

      3. associate-/r/ 98.6%

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

      4. prod-diff 98.6%

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

      5. *-un-lft-identity 98.6%

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

      6. fma-neg 98.6%

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

      7. *-un-lft-identity 98.6%

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

      8. pow1/2 98.6%

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

      9. pow-flip 99.1%

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

      10. metadata-eval 99.1%

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

      11. pow1/2 99.1%

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

      12. pow-flip 99.2%

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

      13. +-commutative 99.2%

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

      14. metadata-eval 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. associate-+l- 99.2%

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

      2. expm1-log1p 99.2%

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

      3. expm1-def 98.6%

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

      4. associate--l- 98.6%

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

      5. fma-udef 98.6%

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

      6. distribute-lft1-in 98.6%

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

      7. metadata-eval 98.6%

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

      8. mul0-lft 98.6%

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

      9. metadata-eval 98.6%

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

      10. expm1-def 99.2%

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

      11. expm1-log1p 99.2%

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

   5. Simplified 99.2%

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

   if 1.42e8 < x < 5.49999999999999981e102

   1. Initial program 5.4%

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

      1. *-un-lft-identity 5.4%

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

      2. clear-num 5.4%

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

      3. associate-/r/ 5.4%

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

      4. prod-diff 5.4%

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

      5. *-un-lft-identity 5.4%

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

      6. fma-neg 5.4%

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

      7. *-un-lft-identity 5.4%

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

      8. pow1/2 5.4%

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

      9. pow-flip 5.5%

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

      10. metadata-eval 5.5%

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

      11. pow1/2 5.5%

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

      12. pow-flip 5.4%

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

      13. +-commutative 5.4%

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

      14. metadata-eval 5.4%

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

   3. Applied egg-rr 5.4%

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

      1. associate-+l- 5.4%

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

      2. expm1-log1p 5.4%

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

      3. expm1-def 6.6%

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

      4. associate--l- 6.6%

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

      5. fma-udef 6.6%

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

      6. distribute-lft1-in 6.6%

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

      7. metadata-eval 6.6%

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

      8. mul0-lft 6.6%

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

      9. metadata-eval 6.6%

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

      10. expm1-def 5.4%

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

      11. expm1-log1p 5.4%

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

   5. Simplified 5.4%

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

   6. Taylor expanded in x around inf 98.8%

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

   if 5.49999999999999981e102 < x

   1. Initial program 47.1%

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

      1. sub-neg 47.1%

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

      2. flip-+ 47.1%

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

      3. frac-times 33.7%

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

      4. metadata-eval 33.7%

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

      5. add-sqr-sqrt 25.0%

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

      6. distribute-neg-frac 25.0%

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

      7. metadata-eval 25.0%

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

      8. +-commutative 25.0%

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

      9. distribute-neg-frac 25.0%

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

      10. metadata-eval 25.0%

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

      11. +-commutative 25.0%

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

      12. pow1/2 25.0%

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

      13. pow-flip 25.0%

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

      14. metadata-eval 25.0%

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

   3. Applied egg-rr 25.0%

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

      1. associate-*r/ 28.0%

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

      2. associate-*l/ 28.0%

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

      3. metadata-eval 28.0%

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

      4. associate-/l/ 28.5%

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

      5. rem-square-sqrt 47.1%

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

      6. sub-neg 47.1%

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

      7. distribute-neg-frac 47.1%

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

      8. metadata-eval 47.1%

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

      9. sub-neg 47.1%

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

      10. distribute-neg-frac 47.1%

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

      11. metadata-eval 47.1%

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

   5. Simplified 47.1%

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

   6. Taylor expanded in x around inf 47.1%

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

      1. sqrt-div 47.1%

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

      2. metadata-eval 47.1%

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

      3. sqrt-pow1 53.2%

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

      4. metadata-eval 53.2%

         \[\leadsto \frac{1}{{x}^{\color{blue}{1.5}}} \]

   8. Applied egg-rr 53.2%

      \[\leadsto \color{blue}{\frac{1}{{x}^{1.5}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 142000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{1.5}}\\ \end{array} \]

Alternative 6: 71.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.45) (+ (+ (pow x -0.5) (* x 0.5)) -1.0) (/ 1.0 (pow x 1.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = 1.0 / pow(x, 1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.45d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = 1.0d0 / (x ** 1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.45) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = 1.0 / Math.pow(x, 1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.45:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = 1.0 / math.pow(x, 1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.45)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(1.0 / (x ^ 1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.45)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = 1.0 / (x ^ 1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.45], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 100.0%

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

      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. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. mul0-lft 100.0%

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

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.8%

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

   if 1.44999999999999996 < x

   1. Initial program 36.2%

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

      1. sub-neg 36.2%

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

      2. flip-+ 36.2%

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

      3. frac-times 27.5%

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

      4. metadata-eval 27.5%

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

      5. add-sqr-sqrt 21.6%

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

      6. distribute-neg-frac 21.6%

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

      7. metadata-eval 21.6%

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

      8. +-commutative 21.6%

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

      9. distribute-neg-frac 21.6%

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

      10. metadata-eval 21.6%

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

      11. +-commutative 21.6%

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

      12. pow1/2 21.6%

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

      13. pow-flip 21.6%

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

      14. metadata-eval 21.6%

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

   3. Applied egg-rr 21.6%

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

      1. associate-*r/ 23.6%

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

      2. associate-*l/ 23.6%

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

      3. metadata-eval 23.6%

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

      4. associate-/l/ 23.8%

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

      5. rem-square-sqrt 36.4%

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

      6. sub-neg 36.4%

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

      7. distribute-neg-frac 36.4%

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

      8. metadata-eval 36.4%

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

      9. sub-neg 36.4%

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

      10. distribute-neg-frac 36.4%

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

      11. metadata-eval 36.4%

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

   5. Simplified 36.4%

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

   6. Taylor expanded in x around inf 37.8%

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

      1. sqrt-div 37.8%

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

      2. metadata-eval 37.8%

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

      3. sqrt-pow1 42.0%

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

      4. metadata-eval 42.0%

         \[\leadsto \frac{1}{{x}^{\color{blue}{1.5}}} \]

   8. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\frac{1}{{x}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{1.5}}\\ \end{array} \]

Alternative 7: 71.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.84:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.84) (+ (pow x -0.5) -1.0) (pow x -1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.84) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.84d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = x ** (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.84) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.84:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.84)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = x ^ -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.84)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = x ^ -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.84], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[Power[x, -1.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.839999999999999969

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 100.0%

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

      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. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. mul0-lft 100.0%

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

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.4%

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

   if 0.839999999999999969 < x

   1. Initial program 36.2%

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

      1. sub-neg 36.2%

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

      2. flip-+ 36.2%

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

      3. frac-times 27.5%

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

      4. metadata-eval 27.5%

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

      5. add-sqr-sqrt 21.6%

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

      6. distribute-neg-frac 21.6%

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

      7. metadata-eval 21.6%

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

      8. +-commutative 21.6%

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

      9. distribute-neg-frac 21.6%

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

      10. metadata-eval 21.6%

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

      11. +-commutative 21.6%

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

      12. pow1/2 21.6%

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

      13. pow-flip 21.6%

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

      14. metadata-eval 21.6%

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

   3. Applied egg-rr 21.6%

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

      1. associate-*r/ 23.6%

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

      2. associate-*l/ 23.6%

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

      3. metadata-eval 23.6%

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

      4. associate-/l/ 23.8%

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

      5. rem-square-sqrt 36.4%

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

      6. sub-neg 36.4%

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

      7. distribute-neg-frac 36.4%

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

      8. metadata-eval 36.4%

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

      9. sub-neg 36.4%

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

      10. distribute-neg-frac 36.4%

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

      11. metadata-eval 36.4%

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

   5. Simplified 36.4%

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

      1. frac-add 38.2%

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

      2. div-inv 38.2%

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

      3. *-un-lft-identity 38.2%

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

      4. associate-+l+ 80.0%

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

      5. metadata-eval 80.0%

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

      6. frac-times 81.6%

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

      7. un-div-inv 81.7%

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

   7. Applied egg-rr 81.7%

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

   8. Taylor expanded in x around inf 37.8%

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

      1. unpow-1 37.8%

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

      2. exp-to-pow 37.8%

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

      3. *-commutative 37.8%

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

      4. exp-prod 37.9%

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

      5. *-commutative 37.9%

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

      6. associate-*l* 37.9%

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

      7. metadata-eval 37.9%

         \[\leadsto \sqrt{e^{\log x \cdot \color{blue}{-3}}} \]

      8. exp-to-pow 37.9%

         \[\leadsto \sqrt{\color{blue}{{x}^{-3}}} \]

      9. metadata-eval 37.9%

         \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

      10. pow-sqr 37.9%

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

      11. rem-sqrt-square 42.0%

          \[\leadsto \color{blue}{\left|{x}^{-1.5}\right|} \]

      12. rem-square-sqrt 42.0%

          \[\leadsto \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

      13. fabs-sqr 42.0%

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

      14. rem-square-sqrt 42.0%

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

   10. Simplified 42.0%

       \[\leadsto \color{blue}{{x}^{-1.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.84:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \]

Alternative 8: 71.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.84:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.84) (+ (pow x -0.5) -1.0) (/ 1.0 (pow x 1.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.84) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / pow(x, 1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.84d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 1.0d0 / (x ** 1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.84) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / Math.pow(x, 1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.84:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 1.0 / math.pow(x, 1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.84)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(1.0 / (x ^ 1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.84)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 1.0 / (x ^ 1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.84], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 / N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.839999999999999969

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 100.0%

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

      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. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. mul0-lft 100.0%

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

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.4%

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

   if 0.839999999999999969 < x

   1. Initial program 36.2%

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

      1. sub-neg 36.2%

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

      2. flip-+ 36.2%

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

      3. frac-times 27.5%

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

      4. metadata-eval 27.5%

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

      5. add-sqr-sqrt 21.6%

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

      6. distribute-neg-frac 21.6%

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

      7. metadata-eval 21.6%

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

      8. +-commutative 21.6%

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

      9. distribute-neg-frac 21.6%

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

      10. metadata-eval 21.6%

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

      11. +-commutative 21.6%

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

      12. pow1/2 21.6%

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

      13. pow-flip 21.6%

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

      14. metadata-eval 21.6%

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

   3. Applied egg-rr 21.6%

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

      1. associate-*r/ 23.6%

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

      2. associate-*l/ 23.6%

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

      3. metadata-eval 23.6%

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

      4. associate-/l/ 23.8%

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

      5. rem-square-sqrt 36.4%

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

      6. sub-neg 36.4%

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

      7. distribute-neg-frac 36.4%

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

      8. metadata-eval 36.4%

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

      9. sub-neg 36.4%

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

      10. distribute-neg-frac 36.4%

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

      11. metadata-eval 36.4%

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

   5. Simplified 36.4%

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

   6. Taylor expanded in x around inf 37.8%

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

      1. sqrt-div 37.8%

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

      2. metadata-eval 37.8%

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

      3. sqrt-pow1 42.0%

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

      4. metadata-eval 42.0%

         \[\leadsto \frac{1}{{x}^{\color{blue}{1.5}}} \]

   8. Applied egg-rr 42.0%

      \[\leadsto \color{blue}{\frac{1}{{x}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.84:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{1.5}}\\ \end{array} \]

Alternative 9: 70.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.0) (pow x -0.5) (pow x -1.5)))

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = pow(x, -0.5);
	} else {
		tmp = pow(x, -1.5);
	}
	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)
    else
        tmp = x ** (-1.5d0)
    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);
	} else {
		tmp = Math.pow(x, -1.5);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.0)
		tmp = x ^ -0.5;
	else
		tmp = x ^ -1.5;
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.0], N[Power[x, -0.5], $MachinePrecision], N[Power[x, -1.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.5%

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

      1. inv-pow 99.6%

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

      2. add-sqr-sqrt 99.2%

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

      3. unpow-prod-down 99.0%

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

      4. pow1/2 99.0%

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

      5. sqrt-pow1 99.1%

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

      6. metadata-eval 99.1%

         \[\leadsto {\left({x}^{\color{blue}{0.25}}\right)}^{-1} \cdot {\left(\sqrt{\sqrt{x}}\right)}^{-1} - \frac{1}{\sqrt{x + 1}} \]

      7. pow1/2 99.1%

         \[\leadsto {\left({x}^{0.25}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{{x}^{0.5}}}\right)}^{-1} - \frac{1}{\sqrt{x + 1}} \]

      8. sqrt-pow1 99.0%

         \[\leadsto {\left({x}^{0.25}\right)}^{-1} \cdot {\color{blue}{\left({x}^{\left(\frac{0.5}{2}\right)}\right)}}^{-1} - \frac{1}{\sqrt{x + 1}} \]

      9. metadata-eval 99.0%

         \[\leadsto {\left({x}^{0.25}\right)}^{-1} \cdot {\left({x}^{\color{blue}{0.25}}\right)}^{-1} - \frac{1}{\sqrt{x + 1}} \]

   3. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{-1} \cdot {\left({x}^{0.25}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}} \]
   4. Step-by-step derivation

      1. pow-sqr 99.3%

         \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{\left(2 \cdot -1\right)}} - \frac{1}{\sqrt{x + 1}} \]

      2. metadata-eval 99.3%

         \[\leadsto {\left({x}^{0.25}\right)}^{\color{blue}{-2}} - \frac{1}{\sqrt{x + 1}} \]

   5. Simplified 99.3%

      \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{-2}} - \frac{1}{\sqrt{x + 1}} \]

   6. Taylor expanded in x around inf 95.9%

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

      1. inv-pow 95.9%

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

      2. sqrt-pow1 96.0%

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

      3. metadata-eval 96.0%

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

      4. expm1-log1p-u 89.0%

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

      5. expm1-udef 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. expm1-def 89.0%

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

      2. expm1-log1p 96.0%

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

   10. Simplified 96.0%

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

   if 1 < x

   1. Initial program 36.2%

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

      1. sub-neg 36.2%

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

      2. flip-+ 36.2%

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

      3. frac-times 27.5%

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

      4. metadata-eval 27.5%

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

      5. add-sqr-sqrt 21.6%

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

      6. distribute-neg-frac 21.6%

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

      7. metadata-eval 21.6%

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

      8. +-commutative 21.6%

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

      9. distribute-neg-frac 21.6%

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

      10. metadata-eval 21.6%

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

      11. +-commutative 21.6%

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

      12. pow1/2 21.6%

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

      13. pow-flip 21.6%

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

      14. metadata-eval 21.6%

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

   3. Applied egg-rr 21.6%

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

      1. associate-*r/ 23.6%

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

      2. associate-*l/ 23.6%

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

      3. metadata-eval 23.6%

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

      4. associate-/l/ 23.8%

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

      5. rem-square-sqrt 36.4%

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

      6. sub-neg 36.4%

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

      7. distribute-neg-frac 36.4%

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

      8. metadata-eval 36.4%

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

      9. sub-neg 36.4%

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

      10. distribute-neg-frac 36.4%

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

      11. metadata-eval 36.4%

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

   5. Simplified 36.4%

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

      1. frac-add 38.2%

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

      2. div-inv 38.2%

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

      3. *-un-lft-identity 38.2%

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

      4. associate-+l+ 80.0%

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

      5. metadata-eval 80.0%

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

      6. frac-times 81.6%

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

      7. un-div-inv 81.7%

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

   7. Applied egg-rr 81.7%

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

   8. Taylor expanded in x around inf 37.8%

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

      1. unpow-1 37.8%

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

      2. exp-to-pow 37.8%

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

      3. *-commutative 37.8%

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

      4. exp-prod 37.9%

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

      5. *-commutative 37.9%

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

      6. associate-*l* 37.9%

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

      7. metadata-eval 37.9%

         \[\leadsto \sqrt{e^{\log x \cdot \color{blue}{-3}}} \]

      8. exp-to-pow 37.9%

         \[\leadsto \sqrt{\color{blue}{{x}^{-3}}} \]

      9. metadata-eval 37.9%

         \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

      10. pow-sqr 37.9%

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

      11. rem-sqrt-square 42.0%

          \[\leadsto \color{blue}{\left|{x}^{-1.5}\right|} \]

      12. rem-square-sqrt 42.0%

          \[\leadsto \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

      13. fabs-sqr 42.0%

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

      14. rem-square-sqrt 42.0%

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

   10. Simplified 42.0%

       \[\leadsto \color{blue}{{x}^{-1.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5}\\ \end{array} \]

Alternative 10: 25.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -1.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -1.5], $MachinePrecision]

Derivation

1. Initial program 67.6%

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

   1. sub-neg 67.6%

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

   2. flip-+ 67.6%

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

   3. frac-times 63.1%

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

   4. metadata-eval 63.1%

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

   5. add-sqr-sqrt 60.1%

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

   6. distribute-neg-frac 60.1%

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

   7. metadata-eval 60.1%

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

   8. +-commutative 60.1%

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

   9. distribute-neg-frac 60.1%

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

   10. metadata-eval 60.1%

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

   11. +-commutative 60.1%

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

   12. pow1/2 60.1%

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

   13. pow-flip 60.1%

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

   14. metadata-eval 60.1%

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

3. Applied egg-rr 60.1%

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

   1. associate-*r/ 61.1%

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

   2. associate-*l/ 61.1%

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

   3. metadata-eval 61.1%

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

   4. associate-/l/ 61.2%

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

   5. rem-square-sqrt 67.5%

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

   6. sub-neg 67.5%

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

   7. distribute-neg-frac 67.5%

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

   8. metadata-eval 67.5%

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

   9. sub-neg 67.5%

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

   10. distribute-neg-frac 67.5%

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

   11. metadata-eval 67.5%

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

5. Simplified 67.5%

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

   1. frac-add 68.5%

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

   2. div-inv 68.5%

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

   3. *-un-lft-identity 68.5%

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

   4. associate-+l+ 89.6%

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

   5. metadata-eval 89.6%

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

   6. frac-times 90.4%

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

   7. un-div-inv 90.4%

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

7. Applied egg-rr 90.4%

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

8. Taylor expanded in x around inf 21.7%

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

   1. unpow-1 21.7%

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

   2. exp-to-pow 21.7%

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

   3. *-commutative 21.7%

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

   4. exp-prod 21.8%

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

   5. *-commutative 21.8%

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

   6. associate-*l* 21.8%

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

   7. metadata-eval 21.8%

      \[\leadsto \sqrt{e^{\log x \cdot \color{blue}{-3}}} \]

   8. exp-to-pow 21.8%

      \[\leadsto \sqrt{\color{blue}{{x}^{-3}}} \]

   9. metadata-eval 21.8%

      \[\leadsto \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

   10. pow-sqr 21.8%

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

   11. rem-sqrt-square 23.9%

       \[\leadsto \color{blue}{\left|{x}^{-1.5}\right|} \]

   12. rem-square-sqrt 23.9%

       \[\leadsto \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

   13. fabs-sqr 23.9%

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

   14. rem-square-sqrt 23.9%

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

10. Simplified 23.9%

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

11. Final simplification 23.9%

    \[\leadsto {x}^{-1.5} \]

Alternative 11: 7.4% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return 1.0 / x;
}

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.6%

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

   1. sub-neg 67.6%

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

   2. flip-+ 67.6%

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

   3. frac-times 63.1%

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

   4. metadata-eval 63.1%

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

   5. add-sqr-sqrt 60.1%

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

   6. distribute-neg-frac 60.1%

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

   7. metadata-eval 60.1%

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

   8. +-commutative 60.1%

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

   9. distribute-neg-frac 60.1%

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

   10. metadata-eval 60.1%

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

   11. +-commutative 60.1%

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

   12. pow1/2 60.1%

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

   13. pow-flip 60.1%

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

   14. metadata-eval 60.1%

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

3. Applied egg-rr 60.1%

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

   1. associate-*r/ 61.1%

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

   2. associate-*l/ 61.1%

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

   3. metadata-eval 61.1%

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

   4. associate-/l/ 61.2%

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

   5. rem-square-sqrt 67.5%

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

   6. sub-neg 67.5%

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

   7. distribute-neg-frac 67.5%

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

   8. metadata-eval 67.5%

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

   9. sub-neg 67.5%

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

   10. distribute-neg-frac 67.5%

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

   11. metadata-eval 67.5%

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

5. Simplified 67.5%

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

6. Taylor expanded in x around 0 52.2%

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

   1. distribute-rgt-in 52.2%

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

   2. *-lft-identity 52.2%

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

   3. pow-plus 52.4%

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

   4. metadata-eval 52.4%

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

8. Simplified 52.4%

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

9. Taylor expanded in x around inf 7.3%

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

10. Final simplification 7.3%

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

Developer target: 98.9% 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 2023320 
(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)))))