2isqrt (example 3.6)

Percentage Accurate: 38.7% → 99.7%

Time: 8.5s

Alternatives: 8

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: 38.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+16}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(1 + x, \sqrt{x}, \sqrt{1 + x} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\frac{1}{x}}}{-2 \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e+16)
   (/ (- (+ 1.0 x) x) (fma (+ 1.0 x) (sqrt x) (* (sqrt (+ 1.0 x)) x)))
   (/ (- (sqrt (/ 1.0 x))) (* -2.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 1e+16) {
		tmp = ((1.0 + x) - x) / fma((1.0 + x), sqrt(x), (sqrt((1.0 + x)) * x));
	} else {
		tmp = -sqrt((1.0 / x)) / (-2.0 * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e+16)
		tmp = Float64(Float64(Float64(1.0 + x) - x) / fma(Float64(1.0 + x), sqrt(x), Float64(sqrt(Float64(1.0 + x)) * x)));
	else
		tmp = Float64(Float64(-sqrt(Float64(1.0 / x))) / Float64(-2.0 * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1e+16], N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[(1.0 + x), $MachinePrecision] * N[Sqrt[x], $MachinePrecision] + N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]) / N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e16

   1. Initial program 64.4%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
   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. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. frac-sub N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-rgt-identity N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

   4. Applied rewrites 65.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

   6. Applied rewrites 99.0%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. sqrt-prod N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. rem-square-sqrt N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. *-commutative N/A

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

   8. Applied rewrites 99.3%

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

   if 1e16 < x

   1. Initial program 35.5%

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

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

      13. metadata-eval N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. neg-mul-1 N/A

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

   4. Applied rewrites 35.5%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 99.7

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

   10. Applied rewrites 99.7%

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

   11. Taylor expanded in x around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 99.7

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

   13. Applied rewrites 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+16}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(1 + x, \sqrt{x}, \sqrt{1 + x} \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\frac{1}{x}}}{-2 \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \frac{\frac{-1}{t\_0}}{\mathsf{fma}\left(t\_0, -\sqrt{x}, -x\right)} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (/ (/ (- 1.0) t_0) (fma t_0 (- (sqrt x)) (- x)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return (-1.0 / t_0) / fma(t_0, -sqrt(x), -x);
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	return Float64(Float64(Float64(-1.0) / t_0) / fma(t_0, Float64(-sqrt(x)), Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 37.8%

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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 40.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. *-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. distribute-rgt-out N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-prod N/A

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

   13. lift-*.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. distribute-neg-in N/A

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

   16. +-commutative N/A

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

9. Applied rewrites 99.6%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. frac-2neg N/A

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

    4. metadata-eval N/A

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

    5. un-div-inv N/A

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

    6. lower-/.f64 N/A

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

    7. lower-neg.f64 99.6

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

    8. lift-+.f64 N/A

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

    9. +-commutative N/A

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

    10. lift-+.f64 99.6

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

11. Applied rewrites 99.6%

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

12. Final simplification 99.6%

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

Alternative 3: 98.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \frac{--1}{\mathsf{fma}\left(t\_0, \sqrt{x}, x\right) \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x)))) (/ (- -1.0) (* (fma t_0 (sqrt x) x) t_0))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	return -(-1.0) / (fma(t_0, sqrt(x), x) * t_0);
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	return Float64(Float64(-(-1.0)) / Float64(fma(t_0, sqrt(x), x) * t_0))
end

Wolfram

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

Derivation

1. Initial program 37.8%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 37.8%

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

6. Applied rewrites 40.5%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 4: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{\sqrt{1 + x}} \cdot 1}{\mathsf{fma}\left(-2, x, -0.5\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (/ -1.0 (sqrt (+ 1.0 x))) 1.0) (fma -2.0 x -0.5)))

C

double code(double x) {
	return ((-1.0 / sqrt((1.0 + x))) * 1.0) / fma(-2.0, x, -0.5);
}

Julia

function code(x)
	return Float64(Float64(Float64(-1.0 / sqrt(Float64(1.0 + x))) * 1.0) / fma(-2.0, x, -0.5))
end

Wolfram

code[x_] := N[(N[(N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / N[(-2.0 * x + -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.8%

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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 40.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-rgt-neg-out N/A

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

   4. *-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. distribute-rgt-out N/A

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

   7. lift-sqrt.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. rem-square-sqrt N/A

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

   10. lift-sqrt.f64 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-prod N/A

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

   13. lift-*.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. distribute-neg-in N/A

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

   16. +-commutative N/A

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

9. Applied rewrites 99.6%

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

10. Taylor expanded in x around inf

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

    1. distribute-rgt-in N/A

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

    2. distribute-rgt-in N/A

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

    3. associate-*l* N/A

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

    4. lft-mult-inverse N/A

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

    5. metadata-eval N/A

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

    6. metadata-eval N/A

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

    9. associate-*r* N/A

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

    10. metadata-eval N/A

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

    11. lower-fma.f64 97.9

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

12. Applied rewrites 97.9%

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

13. Final simplification 97.9%

    \[\leadsto \frac{\frac{-1}{\sqrt{1 + x}} \cdot 1}{\mathsf{fma}\left(-2, x, -0.5\right)} \]
14. Add Preprocessing

Alternative 5: 97.6% accurate, 1.4× 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) / 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 37.8%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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. clear-num N/A

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

   4. lift-/.f64 N/A

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

   5. frac-sub N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f64 N/A

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

4. Applied rewrites 37.8%

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

5. Taylor expanded in x around inf

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

   1. lower-/.f64 96.4

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

7. Applied rewrites 96.4%

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

8. Final simplification 96.4%

   \[\leadsto \frac{\frac{0.5}{x}}{\sqrt{1 + x}} \]
9. Add Preprocessing

Alternative 6: 81.4% 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 37.8%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 82.2%

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

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 80.7%

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

Alternative 7: 37.0% 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 37.8%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]
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 34.3%

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

Alternative 8: 7.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{-2 \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -1.0 (* -2.0 x)))

C

double code(double x) {
	return -1.0 / (-2.0 * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-1.0d0) / ((-2.0d0) * x)
end function

Java

public static double code(double x) {
	return -1.0 / (-2.0 * x);
}

Python

def code(x):
	return -1.0 / (-2.0 * x)

Julia

function code(x)
	return Float64(-1.0 / Float64(-2.0 * x))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / (-2.0 * x);
end

Wolfram

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

Derivation

1. Initial program 37.8%

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

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. neg-mul-1 N/A

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

4. Applied rewrites 40.4%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around inf

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

   1. lower-*.f64 96.4

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

10. Applied rewrites 96.4%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 7.9%

    \[\leadsto \frac{\color{blue}{-1}}{-2 \cdot x} \]
14. Add Preprocessing

Developer Target 1: 38.7% 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 2024268 
(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)))))