2isqrt (example 3.6)

Percentage Accurate: 37.8% → 98.8%

Time: 8.5s

Alternatives: 4

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.8% 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: 98.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{x}}\\ 0.5 \cdot \frac{t\_0}{x} + \left(0.5 \cdot \frac{t\_0}{{x}^{2}}\right) \cdot \left(-0.75 + \frac{0.78125}{x}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 x))))
   (+
    (* 0.5 (/ t_0 x))
    (* (* 0.5 (/ t_0 (pow x 2.0))) (+ -0.75 (/ 0.78125 x))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 / x));
	return (0.5 * (t_0 / x)) + ((0.5 * (t_0 / pow(x, 2.0))) * (-0.75 + (0.78125 / x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sqrt((1.0d0 / x))
    code = (0.5d0 * (t_0 / x)) + ((0.5d0 * (t_0 / (x ** 2.0d0))) * ((-0.75d0) + (0.78125d0 / x)))
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 / x));
	return (0.5 * (t_0 / x)) + ((0.5 * (t_0 / Math.pow(x, 2.0))) * (-0.75 + (0.78125 / x)));
}

Python

def code(x):
	t_0 = math.sqrt((1.0 / x))
	return (0.5 * (t_0 / x)) + ((0.5 * (t_0 / math.pow(x, 2.0))) * (-0.75 + (0.78125 / x)))

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 / x))
	return Float64(Float64(0.5 * Float64(t_0 / x)) + Float64(Float64(0.5 * Float64(t_0 / (x ^ 2.0))) * Float64(-0.75 + Float64(0.78125 / x))))
end

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 / x));
	tmp = (0.5 * (t_0 / x)) + ((0.5 * (t_0 / (x ^ 2.0))) * (-0.75 + (0.78125 / x)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]}, N[(N[(0.5 * N[(t$95$0 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[(t$95$0 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.75 + N[(0.78125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 41.0%

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

   1. add-exp-log 41.0%

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

   2. inv-pow 41.0%

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

   3. sqrt-pow2 30.1%

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

   4. metadata-eval 30.1%

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

   5. inv-pow 30.1%

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

   6. sqrt-pow2 41.0%

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

   7. +-commutative 41.0%

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

   8. metadata-eval 41.0%

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

4. Applied egg-rr 41.0%

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

5. Taylor expanded in x around inf 93.8%

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

6. Taylor expanded in x around inf 93.8%

   \[\leadsto e^{\left(\log \left(--0.5 \cdot \sqrt{x}\right) + \left(\color{blue}{\frac{0.25 + 0.5 \cdot \frac{1}{x}}{x}} + 2 \cdot \log \left(\frac{1}{x}\right)\right)\right) - \frac{1}{x}} \]
7. Step-by-step derivation

   1. associate-*r/ 93.8%

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

   2. metadata-eval 93.8%

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

8. Simplified 93.8%

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

9. Taylor expanded in x around inf 93.9%

   \[\leadsto \color{blue}{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)} + \left(-0.75 \cdot \frac{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}}{x} + 0.78125 \cdot \frac{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}}{{x}^{2}}\right)} \]

11. Simplified 99.3%

    \[\leadsto \color{blue}{0.5 \cdot \frac{\sqrt{\frac{1}{x}}}{x} + \left(0.5 \cdot \frac{\sqrt{\frac{1}{x}}}{{x}^{2}}\right) \cdot \left(-0.75 + \frac{0.78125}{x}\right)} \]
12. Add Preprocessing

Alternative 2: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{-0.375 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* -0.375 (sqrt (/ 1.0 (pow x 3.0)))) (* 0.5 (sqrt (/ 1.0 x)))) x))

C

double code(double x) {
	return ((-0.375 * sqrt((1.0 / pow(x, 3.0)))) + (0.5 * sqrt((1.0 / x)))) / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((-0.375d0) * sqrt((1.0d0 / (x ** 3.0d0)))) + (0.5d0 * sqrt((1.0d0 / x)))) / x
end function

Java

public static double code(double x) {
	return ((-0.375 * Math.sqrt((1.0 / Math.pow(x, 3.0)))) + (0.5 * Math.sqrt((1.0 / x)))) / x;
}

Python

def code(x):
	return ((-0.375 * math.sqrt((1.0 / math.pow(x, 3.0)))) + (0.5 * math.sqrt((1.0 / x)))) / x

Julia

function code(x)
	return Float64(Float64(Float64(-0.375 * sqrt(Float64(1.0 / (x ^ 3.0)))) + Float64(0.5 * sqrt(Float64(1.0 / x)))) / x)
end

MATLAB

function tmp = code(x)
	tmp = ((-0.375 * sqrt((1.0 / (x ^ 3.0)))) + (0.5 * sqrt((1.0 / x)))) / x;
end

Wolfram

code[x_] := N[(N[(N[(-0.375 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 41.0%

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

   1. add-exp-log 41.0%

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

   2. inv-pow 41.0%

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

   3. sqrt-pow2 30.1%

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

   4. metadata-eval 30.1%

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

   5. inv-pow 30.1%

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

   6. sqrt-pow2 41.0%

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

   7. +-commutative 41.0%

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

   8. metadata-eval 41.0%

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

4. Applied egg-rr 41.0%

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

5. Taylor expanded in x around inf 93.8%

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

6. Taylor expanded in x around inf 93.9%

   \[\leadsto \color{blue}{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)} + \left(-0.75 \cdot \frac{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}}{x} + 0.78125 \cdot \frac{e^{\log \left(--0.5 \cdot \sqrt{\frac{1}{x}}\right) + \left(-1 \cdot \log \left(\frac{1}{x}\right) + 2 \cdot \log \left(\frac{1}{x}\right)\right)}}{{x}^{2}}\right)} \]

8. Simplified 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 \cdot \sqrt{\frac{1}{x}}, \frac{1}{x}, \mathsf{fma}\left(\left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot {x}^{-3}, 0.78125, -0.75 \cdot \left(\left(0.5 \cdot \sqrt{\frac{1}{x}}\right) \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

9. Taylor expanded in x around inf 99.2%

   \[\leadsto \color{blue}{\frac{-0.375 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}} \]
10. Add Preprocessing

Alternative 3: 97.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-1.5} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.0%

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

   1. add-exp-log 6.9%

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

   2. log-rec 6.9%

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

   3. pow1/2 6.9%

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

   4. log-pow 6.9%

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

   5. +-commutative 6.9%

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

   6. log1p-define 6.9%

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

4. Applied egg-rr 6.9%

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

5. Taylor expanded in x around inf 6.5%

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

   1. distribute-lft-neg-in 6.5%

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

   2. metadata-eval 6.5%

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

   3. *-commutative 6.5%

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

   4. exp-to-pow 40.7%

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

   5. unpow1/2 40.7%

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

   6. +-commutative 40.7%

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

   7. associate--l+ 94.3%

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

7. Simplified 69.7%

   \[\leadsto \color{blue}{0.5 \cdot \sqrt{{x}^{-3}} + 0} \]
8. Step-by-step derivation

   1. +-rgt-identity 69.7%

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

   2. *-commutative 69.7%

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

   3. sqrt-pow1 98.3%

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

   4. metadata-eval 98.3%

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

9. Applied egg-rr 98.3%

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

10. Final simplification 98.3%

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

Alternative 4: 34.8% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 41.0%

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

   1. add-exp-log 6.9%

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

   2. log-rec 6.9%

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

   3. pow1/2 6.9%

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

   4. log-pow 6.9%

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

   5. +-commutative 6.9%

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

   6. log1p-define 6.9%

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

4. Applied egg-rr 6.9%

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

5. Taylor expanded in x around inf 4.6%

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

   1. distribute-lft-neg-in 4.6%

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

   2. metadata-eval 4.6%

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

   3. *-commutative 4.6%

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

   4. exp-to-pow 38.4%

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

   5. unpow1/2 38.4%

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

   6. +-inverses 38.4%

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

7. Simplified 38.4%

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

Developer Target 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 37.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024170 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

  :alt
  (! :herbie-platform default (- (pow x -1/2) (pow (+ x 1) -1/2)))

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