2isqrt (example 3.6)

Percentage Accurate: 38.3% → 99.7%

Time: 9.5s

Alternatives: 6

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.3% 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, 0.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] / N[(x + N[Sqrt[N[Sqrt[x], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

4. Applied egg-rr 39.7%

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

6. Applied egg-rr 82.4%

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

   1. +-commutative N/A

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

   2. rem-square-sqrt N/A

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

   3. accelerator-lowering-hypot.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 99.7

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

8. Applied egg-rr 99.7%

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

   1. pow1/2 N/A

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

   2. inv-pow N/A

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

   3. pow-pow N/A

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

   4. pow-lowering-pow.f64 N/A

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

   5. +-commutative N/A

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

   6. +-lowering-+.f64 N/A

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

   7. metadata-eval 99.7

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

10. Applied egg-rr 99.7%

    \[\leadsto \frac{\color{blue}{{\left(x + 1\right)}^{-0.5}}}{x + \mathsf{hypot}\left(\sqrt{x}, x\right)} \]
11. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(x + N[Sqrt[N[Sqrt[x], $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.4%

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

4. Applied egg-rr 39.7%

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

6. Applied egg-rr 82.4%

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

   1. +-commutative N/A

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

   2. rem-square-sqrt N/A

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

   3. accelerator-lowering-hypot.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 99.7

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.4%

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

4. Applied egg-rr 39.7%

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

6. Applied egg-rr 82.4%

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

   1. +-commutative N/A

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

   2. distribute-lft1-in N/A

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

   3. +-commutative N/A

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

   4. sqrt-prod N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. +-lowering-+.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 99.6

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.4%

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

4. Applied egg-rr 39.7%

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

   1. associate-/l/ N/A

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

   2. associate--l+ N/A

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

   3. +-inverses N/A

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

   4. metadata-eval N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. sqrt-lowering-sqrt.f64 N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. sqrt-lowering-sqrt.f64 N/A

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

   12. +-lowering-+.f64 82.3

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

6. Applied egg-rr 82.3%

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

   1. +-commutative N/A

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

   2. distribute-lft1-in N/A

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

   3. +-commutative N/A

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

   4. sqrt-prod N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. +-lowering-+.f64 N/A

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

   8. sqrt-lowering-sqrt.f64 98.6

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

8. Applied egg-rr 98.6%

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

   1. *-commutative N/A

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

   2. sqrt-unprod N/A

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

   3. +-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. distribute-rgt-out N/A

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

   6. *-commutative N/A

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

   7. distribute-lft1-in N/A

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

   8. +-commutative N/A

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

   9. sqrt-unprod N/A

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

   10. associate-*r* N/A

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

   11. rem-square-sqrt N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. +-commutative N/A

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

   14. +-lowering-+.f64 N/A

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

   15. sqrt-lowering-sqrt.f64 N/A

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

   16. *-lowering-*.f64 N/A

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

   17. sqrt-lowering-sqrt.f64 N/A

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

   18. +-commutative N/A

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

   19. +-lowering-+.f64 98.6

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

10. Applied egg-rr 98.6%

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

Alternative 5: 83.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.4%

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

4. Applied egg-rr 39.7%

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

   1. associate-/l/ N/A

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

   2. associate--l+ N/A

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

   3. +-inverses N/A

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

   4. metadata-eval N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. sqrt-lowering-sqrt.f64 N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. sqrt-lowering-sqrt.f64 N/A

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

   12. +-lowering-+.f64 82.3

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

6. Applied egg-rr 82.3%

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

7. Final simplification 82.3%

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

Alternative 6: 38.3% 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]

Derivation

1. Initial program 37.4%

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

3. Final simplification 37.4%

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

Developer Target 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 38.3% 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 2024207 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

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

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

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