2isqrt (example 3.6)

Percentage Accurate: 38.9% → 99.6%

Time: 8.0s

Alternatives: 9

Speedup: 1.5×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 38.3%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 42.3%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lift-neg.f64 N/A

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

   18. distribute-lft-neg-out N/A

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

9. Applied rewrites 99.5%

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

11. Applied rewrites 99.5%

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

12. Final simplification 99.5%

    \[\leadsto \frac{\frac{-1}{\sqrt{x}}}{\mathsf{fma}\left(\sqrt{1 + x}, -\sqrt{x}, -1 - x\right)} \]
13. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 38.3%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 42.3%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lift-neg.f64 N/A

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

   18. distribute-lft-neg-out N/A

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

9. Applied rewrites 99.5%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-+.f64 99.5

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

    4. lift-neg.f64 N/A

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

    5. lift-+.f64 N/A

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

    6. distribute-neg-in N/A

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

    7. metadata-eval N/A

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

    8. unsub-neg N/A

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

    9. lower--.f64 99.5

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

11. Applied rewrites 99.5%

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

12. Final simplification 99.5%

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

Alternative 3: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{-\sqrt{{x}^{-1}}}{\mathsf{fma}\left(\frac{0.125}{x \cdot x}, x, \mathsf{fma}\left(-2, x, -1.5\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (- (sqrt (pow x -1.0))) (fma (/ 0.125 (* x x)) x (fma -2.0 x -1.5))))

C

double code(double x) {
	return -sqrt(pow(x, -1.0)) / fma((0.125 / (x * x)), x, fma(-2.0, x, -1.5));
}

Julia

function code(x)
	return Float64(Float64(-sqrt((x ^ -1.0))) / fma(Float64(0.125 / Float64(x * x)), x, fma(-2.0, x, -1.5)))
end

Wolfram

code[x_] := N[((-N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]) / N[(N[(0.125 / N[(x * x), $MachinePrecision]), $MachinePrecision] * x + N[(-2.0 * x + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.3%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 42.3%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-lft-in N/A

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

   9. lift-neg.f64 N/A

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

   10. distribute-lft-neg-in N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lift-neg.f64 N/A

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

   18. distribute-lft-neg-out N/A

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in x around inf

    \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{x \cdot \left(\frac{\frac{1}{8}}{{x}^{2}} - \left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)}} \]
11. Step-by-step derivation

    1. sub-neg N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{x \cdot \color{blue}{\left(\frac{\frac{1}{8}}{{x}^{2}} + \left(\mathsf{neg}\left(\left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)\right)\right)}} \]

    2. distribute-rgt-in N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\frac{\frac{1}{8}}{{x}^{2}} \cdot x + \left(\mathsf{neg}\left(\left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)\right) \cdot x}} \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(\frac{\frac{1}{8}}{{x}^{2}}, x, \left(\mathsf{neg}\left(\left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)\right) \cdot x\right)}} \]

    4. lower-/.f64 N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\color{blue}{\frac{\frac{1}{8}}{{x}^{2}}}, x, \left(\mathsf{neg}\left(\left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)\right) \cdot x\right)} \]

    5. unpow2 N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{\color{blue}{x \cdot x}}, x, \left(\mathsf{neg}\left(\left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)\right) \cdot x\right)} \]

    6. lower-*.f64 N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{\color{blue}{x \cdot x}}, x, \left(\mathsf{neg}\left(\left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)\right) \cdot x\right)} \]

    7. distribute-lft-neg-in N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \color{blue}{\mathsf{neg}\left(\left(2 + \frac{3}{2} \cdot \frac{1}{x}\right) \cdot x\right)}\right)} \]

    8. *-commutative N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \mathsf{neg}\left(\color{blue}{x \cdot \left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)}\right)\right)} \]

    9. mul-1-neg N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \color{blue}{-1 \cdot \left(x \cdot \left(2 + \frac{3}{2} \cdot \frac{1}{x}\right)\right)}\right)} \]

    10. distribute-lft-in N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, -1 \cdot \color{blue}{\left(x \cdot 2 + x \cdot \left(\frac{3}{2} \cdot \frac{1}{x}\right)\right)}\right)} \]

    11. distribute-rgt-in N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \color{blue}{\left(x \cdot 2\right) \cdot -1 + \left(x \cdot \left(\frac{3}{2} \cdot \frac{1}{x}\right)\right) \cdot -1}\right)} \]

    12. *-commutative N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \left(x \cdot 2\right) \cdot -1 + \left(x \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{3}{2}\right)}\right) \cdot -1\right)} \]

    13. associate-*r* N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \left(x \cdot 2\right) \cdot -1 + \color{blue}{\left(\left(x \cdot \frac{1}{x}\right) \cdot \frac{3}{2}\right)} \cdot -1\right)} \]

    14. rgt-mult-inverse N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \left(x \cdot 2\right) \cdot -1 + \left(\color{blue}{1} \cdot \frac{3}{2}\right) \cdot -1\right)} \]

    15. metadata-eval N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \left(x \cdot 2\right) \cdot -1 + \color{blue}{\frac{3}{2}} \cdot -1\right)} \]

    16. metadata-eval N/A

        \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(\frac{\frac{1}{8}}{x \cdot x}, x, \left(x \cdot 2\right) \cdot -1 + \color{blue}{\frac{-3}{2}}\right)} \]

12. Applied rewrites 99.0%

    \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(\frac{0.125}{x \cdot x}, x, \mathsf{fma}\left(-2, x, -1.5\right)\right)}} \]

13. Final simplification 99.0%

    \[\leadsto \frac{-\sqrt{{x}^{-1}}}{\mathsf{fma}\left(\frac{0.125}{x \cdot x}, x, \mathsf{fma}\left(-2, x, -1.5\right)\right)} \]
14. Add Preprocessing

Alternative 4: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{-\sqrt{{x}^{-1}}}{\mathsf{fma}\left(-2, x, -1.5\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (sqrt (pow x -1.0))) (fma -2.0 x -1.5)))

C

double code(double x) {
	return -sqrt(pow(x, -1.0)) / fma(-2.0, x, -1.5);
}

Julia

function code(x)
	return Float64(Float64(-sqrt((x ^ -1.0))) / fma(-2.0, x, -1.5))
end

Wolfram

code[x_] := N[((-N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]) / N[(-2.0 * x + -1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.3%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 42.3%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

8. Taylor expanded in x around inf

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

   1. distribute-rgt-in N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. unpow2 N/A

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

   6. associate-*r* N/A

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

   7. lft-mult-inverse N/A

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

   8. *-lft-identity N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. neg-mul-1 N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{-2 \cdot x + \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}\right)\right)}} \]

   14. lower-fma.f64 N/A

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

   15. associate-*l* N/A

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

   16. lft-mult-inverse N/A

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

   17. metadata-eval N/A

       \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\mathsf{fma}\left(-2, x, \mathsf{neg}\left(\color{blue}{\frac{3}{2}}\right)\right)} \]

10. Applied rewrites 98.6%

    \[\leadsto \frac{-\sqrt{\frac{1}{x}}}{\color{blue}{\mathsf{fma}\left(-2, x, -1.5\right)}} \]

11. Final simplification 98.6%

    \[\leadsto \frac{-\sqrt{{x}^{-1}}}{\mathsf{fma}\left(-2, x, -1.5\right)} \]
12. Add Preprocessing

Alternative 5: 5.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (sqrt (pow x -1.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sqrt((x ^ -1.0))
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 38.3%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.8

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

5. Applied rewrites 5.8%

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

6. Final simplification 5.8%

   \[\leadsto \sqrt{{x}^{-1}} \]
7. Add Preprocessing

Alternative 6: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.125}{x} - 0.5}{x}}{-\sqrt{1 + x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ (- (/ 0.125 x) 0.5) x) (- (sqrt (+ 1.0 x)))))

C

double code(double x) {
	return (((0.125 / x) - 0.5) / x) / -sqrt((1.0 + x));
}

Fortran

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

Java

public static double code(double x) {
	return (((0.125 / x) - 0.5) / x) / -Math.sqrt((1.0 + x));
}

Python

def code(x):
	return (((0.125 / x) - 0.5) / x) / -math.sqrt((1.0 + x))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.125 / x) - 0.5) / x) / Float64(-sqrt(Float64(1.0 + x))))
end

MATLAB

function tmp = code(x)
	tmp = (((0.125 / x) - 0.5) / x) / -sqrt((1.0 + x));
end

Wolfram

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

Derivation

1. Initial program 38.3%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 42.3%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 99.3%

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

10. Taylor expanded in x around inf

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8} \cdot \frac{1}{x} - \frac{1}{2}}{x}}}{-\sqrt{1 + x}} \]
11. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{8} \cdot 1}{x}} - \frac{1}{2}}{x}}{-\sqrt{1 + x}} \]

    2. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{\color{blue}{\frac{1}{8}}}{x} - \frac{1}{2}}{x}}{-\sqrt{1 + x}} \]

    3. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{1}{8}}{x} - \frac{1}{2}}{x}}}{-\sqrt{1 + x}} \]

    4. lower--.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{\frac{1}{8}}{x} - \frac{1}{2}}}{x}}{-\sqrt{1 + x}} \]

    5. lower-/.f64 98.7

       \[\leadsto \frac{\frac{\color{blue}{\frac{0.125}{x}} - 0.5}{x}}{-\sqrt{1 + x}} \]

12. Applied rewrites 98.7%

    \[\leadsto \frac{\color{blue}{\frac{\frac{0.125}{x} - 0.5}{x}}}{-\sqrt{1 + x}} \]
13. Add Preprocessing

Alternative 7: 97.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.3%

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

   1. lift--.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-sub N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. *-lft-identity N/A

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

   9. flip-- N/A

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

   10. metadata-eval N/A

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

   11. frac-times N/A

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

   12. frac-2neg N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*r/ N/A

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

4. Applied rewrites 42.3%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-/.f64 99.3

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

7. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

9. Applied rewrites 99.3%

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

10. Taylor expanded in x around inf

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

    1. lower-/.f64 97.3

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

12. Applied rewrites 97.3%

    \[\leadsto \frac{\color{blue}{\frac{-0.5}{x}}}{-\sqrt{1 + x}} \]
13. Add Preprocessing

Alternative 8: 81.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.5 \cdot \sqrt{x}}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* 0.5 (sqrt x)) (* x x)))

C

double code(double x) {
	return (0.5 * sqrt(x)) / (x * x);
}

Fortran

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

Java

public static double code(double x) {
	return (0.5 * Math.sqrt(x)) / (x * x);
}

Python

def code(x):
	return (0.5 * math.sqrt(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(0.5 * sqrt(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 * sqrt(x)) / (x * x);
end

Wolfram

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

Derivation

1. Initial program 38.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \left(\sqrt{\frac{1}{{x}^{3}}} \cdot \left(1 + \frac{1}{4} \cdot x\right)\right) - \left(\frac{-1}{2} \cdot \sqrt{x} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}{{x}^{2}}} \]

5. Applied rewrites 82.9%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(\sqrt{\frac{1}{x}} - \mathsf{fma}\left(\mathsf{fma}\left(0.25, x, 1\right), \sqrt{\frac{1}{{x}^{3}}}, \sqrt{x}\right)\right)}{x \cdot x}} \]

6. Taylor expanded in x around inf

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

8. Applied rewrites 81.4%

   \[\leadsto \frac{0.5 \cdot \sqrt{x}}{\color{blue}{x} \cdot x} \]
9. Add Preprocessing

Alternative 9: 37.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.3%

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

3. Taylor expanded in x around 0

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

   1. lower-sqrt.f64 N/A

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

   2. lower-/.f64 5.8

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

5. Applied rewrites 5.8%

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

7. Applied rewrites 36.0%

   \[\leadsto \sqrt{\frac{x}{x \cdot x}} \]
8. Add Preprocessing

Developer Target 1: 38.9% accurate, 0.2× 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 2024296 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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