2isqrt (example 3.6)

Percentage Accurate: 69.0% → 99.7%

Time: 12.2s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.0% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 0:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 0.0)
     (* (pow x -1.5) 0.5)
     (/ (/ 1.0 (+ (sqrt x) t_0)) (sqrt (fma x x x))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 0.0) {
		tmp = pow(x, -1.5) * 0.5;
	} else {
		tmp = (1.0 / (sqrt(x) + t_0)) / sqrt(fma(x, x, x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 0.0)
		tmp = Float64((x ^ -1.5) * 0.5);
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) / sqrt(fma(x, x, x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 0.0

   1. Initial program 38.4%

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

      1. flip-- 38.4%

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

      2. clear-num 38.4%

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

      3. *-un-lft-identity 38.4%

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

      4. associate-/l* 38.4%

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

      5. flip-- 38.4%

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

      6. pow1/2 38.4%

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

      7. pow-flip 27.0%

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

      8. metadata-eval 27.0%

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

      9. pow1/2 27.0%

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

      10. pow-flip 38.4%

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

      11. +-commutative 38.4%

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

      12. metadata-eval 38.4%

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

   3. Applied egg-rr 38.4%

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

   4. Taylor expanded in x around inf 65.4%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
   5. Step-by-step derivation

      1. div-inv 65.4%

         \[\leadsto \color{blue}{1 \cdot \frac{1}{2 \cdot \sqrt{{x}^{3}}}} \]

      2. associate-/r* 65.4%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}} \]

      3. metadata-eval 65.4%

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

      4. sqrt-pow1 97.7%

         \[\leadsto 1 \cdot \frac{0.5}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \]

      5. metadata-eval 97.7%

         \[\leadsto 1 \cdot \frac{0.5}{{x}^{\color{blue}{1.5}}} \]

   6. Applied egg-rr 97.7%

      \[\leadsto \color{blue}{1 \cdot \frac{0.5}{{x}^{1.5}}} \]
   7. Step-by-step derivation

      1. *-lft-identity 97.7%

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

   8. Simplified 97.7%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
   9. Step-by-step derivation

      1. div-inv 97.7%

         \[\leadsto \color{blue}{0.5 \cdot \frac{1}{{x}^{1.5}}} \]

      2. *-commutative 97.7%

         \[\leadsto \color{blue}{\frac{1}{{x}^{1.5}} \cdot 0.5} \]

      3. pow-flip 100.0%

         \[\leadsto \color{blue}{{x}^{\left(-1.5\right)}} \cdot 0.5 \]

      4. metadata-eval 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 0.0 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 97.5%

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

      1. frac-sub 97.5%

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

      2. *-un-lft-identity 97.5%

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

      3. *-rgt-identity 97.5%

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

      4. flip-- 98.0%

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

      5. associate-/l/ 97.9%

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

      6. add-sqr-sqrt 98.7%

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

      7. +-commutative 98.7%

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

      8. add-sqr-sqrt 99.5%

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

      9. associate--l+ 99.5%

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

      10. sqrt-unprod 99.6%

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

      11. +-commutative 99.6%

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

      12. distribute-rgt-in 99.6%

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

      13. *-un-lft-identity 99.6%

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

      14. pow2 99.6%

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

      15. +-commutative 99.6%

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

      16. +-commutative 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. +-inverses 99.6%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x + {x}^{2}} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]

      2. metadata-eval 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/r* 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. unpow2 99.6%

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

      8. fma-def 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 0:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{\sqrt{\mathsf{fma}\left(x, x, x\right)}}\\ \end{array} \]

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{x + \left(0.5 - \frac{0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 1e-6)
     (/ (/ 1.0 (+ (sqrt x) t_0)) (+ x (- 0.5 (/ 0.125 x))))
     (- (pow x -0.5) (pow (+ 1.0 x) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 1e-6) {
		tmp = (1.0 / (sqrt(x) + t_0)) / (x + (0.5 - (0.125 / x)));
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 1d-6) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) / (x + (0.5d0 - (0.125d0 / x)))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 1e-6) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) / (x + (0.5 - (0.125 / x)));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 1e-6:
		tmp = (1.0 / (math.sqrt(x) + t_0)) / (x + (0.5 - (0.125 / x)))
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 1e-6)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) / Float64(x + Float64(0.5 - Float64(0.125 / x))));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 1e-6)
		tmp = (1.0 / (sqrt(x) + t_0)) / (x + (0.5 - (0.125 / x)));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + N[(0.5 - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 9.99999999999999955e-7

   1. Initial program 39.0%

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

      1. frac-sub 39.0%

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

      2. *-un-lft-identity 39.0%

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

      3. *-rgt-identity 39.0%

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

      4. flip-- 39.5%

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

      5. associate-/l/ 39.6%

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

      6. add-sqr-sqrt 40.2%

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

      7. +-commutative 40.2%

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

      8. add-sqr-sqrt 41.2%

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

      9. associate--l+ 97.3%

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

      10. sqrt-unprod 84.4%

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

      11. +-commutative 84.4%

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

      12. distribute-rgt-in 84.4%

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

      13. *-un-lft-identity 84.4%

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

      14. pow2 84.4%

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

      15. +-commutative 84.4%

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

      16. +-commutative 84.4%

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

   3. Applied egg-rr 84.4%

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

      1. +-inverses 84.4%

         \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x + {x}^{2}} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]

      2. metadata-eval 84.4%

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

      3. *-commutative 84.4%

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

      4. associate-/r* 84.3%

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

      5. +-commutative 84.3%

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

      6. +-commutative 84.3%

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

      7. unpow2 84.3%

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

      8. fma-def 84.3%

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

   5. Simplified 84.3%

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

   6. Taylor expanded in x around inf 99.5%

      \[\leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{\left(0.5 + x\right) - 0.125 \cdot \frac{1}{x}}} \]
   7. Step-by-step derivation

      1. +-commutative 99.5%

         \[\leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{\left(x + 0.5\right)} - 0.125 \cdot \frac{1}{x}} \]

      2. associate--l+ 99.5%

         \[\leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{\color{blue}{x + \left(0.5 - 0.125 \cdot \frac{1}{x}\right)}} \]

      3. associate-*r/ 99.5%

         \[\leadsto \frac{\frac{1}{\sqrt{1 + x} + \sqrt{x}}}{x + \left(0.5 - \color{blue}{\frac{0.125 \cdot 1}{x}}\right)} \]

      4. metadata-eval 99.5%

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

   8. Simplified 99.5%

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

   if 9.99999999999999955e-7 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.7%

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

      1. expm1-log1p-u 99.7%

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

      2. expm1-udef 99.6%

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

      3. associate--r- 99.6%

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

      4. pow1/2 99.7%

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

      5. pow-flip 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow1/2 100.0%

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

      8. pow-flip 100.0%

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

      9. +-commutative 100.0%

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

      10. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-def 100.0%

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

      3. expm1-log1p 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x + \left(0.5 - \frac{0.125}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 10^{-20}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 (sqrt (+ 1.0 x)))) 1e-20)
   (* (pow x -1.5) 0.5)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 1e-20) {
		tmp = pow(x, -1.5) * 0.5;
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) - (1.0d0 / sqrt((1.0d0 + x)))) <= 1d-20) then
        tmp = (x ** (-1.5d0)) * 0.5d0
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / Math.sqrt((1.0 + x)))) <= 1e-20) {
		tmp = Math.pow(x, -1.5) * 0.5;
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / math.sqrt((1.0 + x)))) <= 1e-20:
		tmp = math.pow(x, -1.5) * 0.5
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / sqrt(Float64(1.0 + x)))) <= 1e-20)
		tmp = Float64((x ^ -1.5) * 0.5);
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / sqrt((1.0 + x)))) <= 1e-20)
		tmp = (x ^ -1.5) * 0.5;
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-20], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 9.99999999999999945e-21

   1. Initial program 38.1%

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

      1. flip-- 38.1%

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

      2. clear-num 38.1%

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

      3. *-un-lft-identity 38.1%

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

      4. associate-/l* 38.1%

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

      5. flip-- 38.1%

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

      6. pow1/2 38.1%

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

      7. pow-flip 27.0%

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

      8. metadata-eval 27.0%

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

      9. pow1/2 27.0%

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

      10. pow-flip 38.2%

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

      11. +-commutative 38.2%

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

      12. metadata-eval 38.2%

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

   3. Applied egg-rr 38.2%

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

   4. Taylor expanded in x around inf 66.1%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
   5. Step-by-step derivation

      1. div-inv 66.1%

         \[\leadsto \color{blue}{1 \cdot \frac{1}{2 \cdot \sqrt{{x}^{3}}}} \]

      2. associate-/r* 66.1%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}} \]

      3. metadata-eval 66.1%

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

      4. sqrt-pow1 97.6%

         \[\leadsto 1 \cdot \frac{0.5}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \]

      5. metadata-eval 97.6%

         \[\leadsto 1 \cdot \frac{0.5}{{x}^{\color{blue}{1.5}}} \]

   6. Applied egg-rr 97.6%

      \[\leadsto \color{blue}{1 \cdot \frac{0.5}{{x}^{1.5}}} \]
   7. Step-by-step derivation

      1. *-lft-identity 97.6%

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

   8. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
   9. Step-by-step derivation

      1. div-inv 97.6%

         \[\leadsto \color{blue}{0.5 \cdot \frac{1}{{x}^{1.5}}} \]

      2. *-commutative 97.6%

         \[\leadsto \color{blue}{\frac{1}{{x}^{1.5}} \cdot 0.5} \]

      3. pow-flip 99.8%

         \[\leadsto \color{blue}{{x}^{\left(-1.5\right)}} \cdot 0.5 \]

      4. metadata-eval 99.8%

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

   10. Applied egg-rr 99.8%

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

   if 9.99999999999999945e-21 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.1%

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

      1. expm1-log1p-u 99.1%

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

      2. expm1-udef 98.8%

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

      3. associate--r- 98.8%

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

      4. pow1/2 98.8%

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

      5. pow-flip 99.1%

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

      6. metadata-eval 99.1%

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

      7. pow1/2 99.1%

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

      8. pow-flip 99.1%

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

      9. +-commutative 99.1%

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

      10. metadata-eval 99.1%

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

   3. Applied egg-rr 99.1%

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

      1. associate-+l- 99.1%

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

      2. expm1-def 99.5%

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

      3. expm1-log1p 99.5%

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

   5. Simplified 99.5%

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{1 + x}} \leq 10^{-20}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.7%

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

      1. flip-- 99.5%

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

      2. clear-num 99.4%

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

      3. *-un-lft-identity 99.4%

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

      4. associate-/l* 99.5%

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

      5. flip-- 99.6%

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

      6. pow1/2 99.6%

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

      7. pow-flip 99.9%

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

      8. metadata-eval 99.9%

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

      9. pow1/2 99.9%

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

      10. pow-flip 99.9%

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

      11. +-commutative 99.9%

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

      12. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 98.8%

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

   5. Taylor expanded in x around 0 98.9%

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

   if 1 < x

   1. Initial program 39.0%

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

      1. flip-- 39.0%

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

      2. clear-num 39.0%

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

      3. *-un-lft-identity 39.0%

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

      4. associate-/l* 39.0%

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

      5. flip-- 39.0%

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

      6. pow1/2 39.0%

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

      7. pow-flip 28.1%

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

      8. metadata-eval 28.1%

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

      9. pow1/2 28.1%

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

      10. pow-flip 39.1%

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

      11. +-commutative 39.1%

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

      12. metadata-eval 39.1%

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

   3. Applied egg-rr 39.1%

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

   4. Taylor expanded in x around inf 65.5%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
   5. Step-by-step derivation

      1. div-inv 65.5%

         \[\leadsto \color{blue}{1 \cdot \frac{1}{2 \cdot \sqrt{{x}^{3}}}} \]

      2. associate-/r* 65.5%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}} \]

      3. metadata-eval 65.5%

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

      4. sqrt-pow1 96.3%

         \[\leadsto 1 \cdot \frac{0.5}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \]

      5. metadata-eval 96.3%

         \[\leadsto 1 \cdot \frac{0.5}{{x}^{\color{blue}{1.5}}} \]

   6. Applied egg-rr 96.3%

      \[\leadsto \color{blue}{1 \cdot \frac{0.5}{{x}^{1.5}}} \]
   7. Step-by-step derivation

      1. *-lft-identity 96.3%

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

   8. Simplified 96.3%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
   9. Step-by-step derivation

      1. div-inv 96.3%

         \[\leadsto \color{blue}{0.5 \cdot \frac{1}{{x}^{1.5}}} \]

      2. *-commutative 96.3%

         \[\leadsto \color{blue}{\frac{1}{{x}^{1.5}} \cdot 0.5} \]

      3. pow-flip 98.5%

         \[\leadsto \color{blue}{{x}^{\left(-1.5\right)}} \cdot 0.5 \]

      4. metadata-eval 98.5%

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

   10. Applied egg-rr 98.5%

       \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

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

Alternative 5: 98.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.66) (+ (pow x -0.5) -1.0) (* (pow x -1.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.66) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.66d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (x ** (-1.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.66) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = Math.pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.66:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = math.pow(x, -1.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64((x ^ -1.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.66)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (x ^ -1.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.66], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.660000000000000031

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.1%

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

      1. expm1-log1p-u 92.2%

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

      2. expm1-udef 92.2%

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

      3. pow1/2 92.2%

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

      4. pow-flip 92.2%

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

      5. metadata-eval 92.2%

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

   4. Applied egg-rr 92.2%

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

      1. expm1-def 92.2%

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

      2. expm1-log1p 99.4%

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

   6. Simplified 99.4%

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

   if 0.660000000000000031 < x

   1. Initial program 39.5%

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

      1. flip-- 39.5%

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

      2. clear-num 39.5%

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

      3. *-un-lft-identity 39.5%

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

      4. associate-/l* 39.5%

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

      5. flip-- 39.5%

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

      6. pow1/2 39.5%

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

      7. pow-flip 28.7%

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

      8. metadata-eval 28.7%

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

      9. pow1/2 28.7%

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

      10. pow-flip 39.5%

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

      11. +-commutative 39.5%

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

      12. metadata-eval 39.5%

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

   3. Applied egg-rr 39.5%

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

   4. Taylor expanded in x around inf 65.2%

      \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
   5. Step-by-step derivation

      1. div-inv 65.2%

         \[\leadsto \color{blue}{1 \cdot \frac{1}{2 \cdot \sqrt{{x}^{3}}}} \]

      2. associate-/r* 65.2%

         \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}} \]

      3. metadata-eval 65.2%

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

      4. sqrt-pow1 95.7%

         \[\leadsto 1 \cdot \frac{0.5}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \]

      5. metadata-eval 95.7%

         \[\leadsto 1 \cdot \frac{0.5}{{x}^{\color{blue}{1.5}}} \]

   6. Applied egg-rr 95.7%

      \[\leadsto \color{blue}{1 \cdot \frac{0.5}{{x}^{1.5}}} \]
   7. Step-by-step derivation

      1. *-lft-identity 95.7%

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

   8. Simplified 95.7%

      \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
   9. Step-by-step derivation

      1. div-inv 95.7%

         \[\leadsto \color{blue}{0.5 \cdot \frac{1}{{x}^{1.5}}} \]

      2. *-commutative 95.7%

         \[\leadsto \color{blue}{\frac{1}{{x}^{1.5}} \cdot 0.5} \]

      3. pow-flip 97.9%

         \[\leadsto \color{blue}{{x}^{\left(-1.5\right)}} \cdot 0.5 \]

      4. metadata-eval 97.9%

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

   10. Applied egg-rr 97.9%

       \[\leadsto \color{blue}{{x}^{-1.5} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]

Alternative 6: 51.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. flip-- 68.8%

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

   2. clear-num 68.7%

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

   3. *-un-lft-identity 68.7%

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

   4. associate-/l* 68.8%

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

   5. flip-- 68.9%

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

   6. pow1/2 68.9%

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

   7. pow-flip 63.4%

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

   8. metadata-eval 63.4%

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

   9. pow1/2 63.4%

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

   10. pow-flip 69.0%

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

   11. +-commutative 69.0%

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

   12. metadata-eval 69.0%

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

3. Applied egg-rr 69.0%

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

4. Taylor expanded in x around inf 36.0%

   \[\leadsto \frac{1}{\color{blue}{2 \cdot \sqrt{{x}^{3}}}} \]
5. Step-by-step derivation

   1. div-inv 36.0%

      \[\leadsto \color{blue}{1 \cdot \frac{1}{2 \cdot \sqrt{{x}^{3}}}} \]

   2. associate-/r* 36.0%

      \[\leadsto 1 \cdot \color{blue}{\frac{\frac{1}{2}}{\sqrt{{x}^{3}}}} \]

   3. metadata-eval 36.0%

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

   4. sqrt-pow1 51.7%

      \[\leadsto 1 \cdot \frac{0.5}{\color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \]

   5. metadata-eval 51.7%

      \[\leadsto 1 \cdot \frac{0.5}{{x}^{\color{blue}{1.5}}} \]

6. Applied egg-rr 51.7%

   \[\leadsto \color{blue}{1 \cdot \frac{0.5}{{x}^{1.5}}} \]
7. Step-by-step derivation

   1. *-lft-identity 51.7%

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

8. Simplified 51.7%

   \[\leadsto \color{blue}{\frac{0.5}{{x}^{1.5}}} \]
9. Step-by-step derivation

   1. div-inv 51.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{1}{{x}^{1.5}}} \]

   2. *-commutative 51.7%

      \[\leadsto \color{blue}{\frac{1}{{x}^{1.5}} \cdot 0.5} \]

   3. pow-flip 52.9%

      \[\leadsto \color{blue}{{x}^{\left(-1.5\right)}} \cdot 0.5 \]

   4. metadata-eval 52.9%

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

10. Applied egg-rr 52.9%

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

11. Final simplification 52.9%

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

Alternative 7: 7.4% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. frac-sub 68.9%

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

   2. *-un-lft-identity 68.9%

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

   3. *-rgt-identity 68.9%

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

   4. flip-- 69.1%

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

   5. associate-/l/ 69.1%

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

   6. add-sqr-sqrt 69.4%

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

   7. +-commutative 69.4%

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

   8. add-sqr-sqrt 69.9%

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

   9. associate--l+ 98.4%

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

   10. sqrt-unprod 91.9%

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

   11. +-commutative 91.9%

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

   12. distribute-rgt-in 91.9%

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

   13. *-un-lft-identity 91.9%

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

   14. pow2 91.9%

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

   15. +-commutative 91.9%

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

   16. +-commutative 91.9%

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

3. Applied egg-rr 91.9%

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

   1. +-inverses 91.9%

      \[\leadsto \frac{1 + \color{blue}{0}}{\sqrt{x + {x}^{2}} \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)} \]

   2. metadata-eval 91.9%

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

   3. *-commutative 91.9%

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

   4. associate-/r* 91.9%

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

   5. +-commutative 91.9%

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

   6. +-commutative 91.9%

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

   7. unpow2 91.9%

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

   8. fma-def 91.9%

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

5. Simplified 91.9%

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

6. Taylor expanded in x around inf 53.4%

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

7. Taylor expanded in x around 0 7.3%

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

8. Final simplification 7.3%

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

Alternative 8: 1.9% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 68.9%

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

2. Taylor expanded in x around 0 49.8%

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

3. Taylor expanded in x around inf 2.0%

   \[\leadsto \color{blue}{-1} \]

4. Final simplification 2.0%

   \[\leadsto -1 \]

Alternative 9: 5.8% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 68.9%

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

   1. frac-sub 68.9%

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

   2. div-inv 68.9%

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

   3. *-un-lft-identity 68.9%

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

   4. *-rgt-identity 68.9%

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

   5. flip-- 69.1%

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

   6. metadata-eval 69.1%

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

   7. frac-times 69.1%

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

   8. associate-*l/ 69.1%

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

3. Applied egg-rr 91.9%

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

4. Taylor expanded in x around inf 53.8%

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

6. Simplified 53.8%

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

7. Taylor expanded in x around 0 5.7%

   \[\leadsto \color{blue}{2} \]

8. Final simplification 5.7%

   \[\leadsto 2 \]

Developer target: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023301 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ (* (+ x 1.0) (sqrt x)) (* x (sqrt (+ x 1.0)))))

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