2isqrt (example 3.6)

Percentage Accurate: 68.6% → 99.6%

Time: 10.6s

Alternatives: 16

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: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 10^{-10}:\\ \;\;\;\;{\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \frac{\sqrt{x}}{t_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 1e-10)
     (* (pow (+ 1.0 x) -0.5) (/ 1.0 (- (+ 0.5 (* x 2.0)) (/ 0.125 x))))
     (* (pow x -0.5) (- 1.0 (/ (sqrt x) t_0))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 1e-10) {
		tmp = pow((1.0 + x), -0.5) * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)));
	} else {
		tmp = pow(x, -0.5) * (1.0 - (sqrt(x) / t_0));
	}
	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-10) then
        tmp = ((1.0d0 + x) ** (-0.5d0)) * (1.0d0 / ((0.5d0 + (x * 2.0d0)) - (0.125d0 / x)))
    else
        tmp = (x ** (-0.5d0)) * (1.0d0 - (sqrt(x) / t_0))
    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-10) {
		tmp = Math.pow((1.0 + x), -0.5) * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)));
	} else {
		tmp = Math.pow(x, -0.5) * (1.0 - (Math.sqrt(x) / t_0));
	}
	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-10:
		tmp = math.pow((1.0 + x), -0.5) * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)))
	else:
		tmp = math.pow(x, -0.5) * (1.0 - (math.sqrt(x) / t_0))
	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-10)
		tmp = Float64((Float64(1.0 + x) ^ -0.5) * Float64(1.0 / Float64(Float64(0.5 + Float64(x * 2.0)) - Float64(0.125 / x))));
	else
		tmp = Float64((x ^ -0.5) * Float64(1.0 - Float64(sqrt(x) / t_0)));
	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-10)
		tmp = ((1.0 + x) ^ -0.5) * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)));
	else
		tmp = (x ^ -0.5) * (1.0 - (sqrt(x) / t_0));
	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-10], N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 / N[(N[(0.5 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(1.0 - N[(N[Sqrt[x], $MachinePrecision] / t$95$0), $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)))) < 1.00000000000000004e-10

   1. Initial program 41.9%

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

      1. frac-sub 42.0%

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

      2. associate-/r* 42.0%

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

      3. *-un-lft-identity 42.0%

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

      4. *-rgt-identity 42.0%

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

      5. +-commutative 42.0%

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

      6. +-commutative 42.0%

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

   3. Applied egg-rr 42.0%

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

      1. flip-- 42.7%

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

      2. div-inv 42.7%

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

      3. add-sqr-sqrt 24.7%

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

      4. +-commutative 24.7%

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

      5. add-sqr-sqrt 44.4%

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

      6. associate--l+ 44.4%

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

      7. +-commutative 44.4%

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

      8. +-commutative 44.4%

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

   5. Applied egg-rr 44.4%

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

      1. associate-*r/ 44.4%

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

      2. *-rgt-identity 44.4%

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

      3. associate-+r- 44.4%

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

      4. +-commutative 44.4%

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

      5. associate--l+ 99.3%

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

   7. Simplified 99.3%

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

      1. clear-num 98.7%

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

      2. +-commutative 98.7%

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

      3. associate-/r/ 99.3%

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

      4. inv-pow 99.3%

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

      5. sqrt-pow2 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-/l/ 99.3%

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

      9. +-inverses 99.3%

         \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1 + \color{blue}{0}}{\sqrt{x} \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]

      10. metadata-eval 99.3%

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

      11. distribute-lft-in 99.3%

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

      12. add-sqr-sqrt 99.5%

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

      13. sqrt-unprod 82.9%

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

      14. +-commutative 82.9%

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

   9. Applied egg-rr 82.9%

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

   10. Taylor expanded in x around inf 99.8%

       \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\color{blue}{\left(0.5 + 2 \cdot x\right) - 0.125 \cdot \frac{1}{x}}} \]
   11. Step-by-step derivation

       1. *-commutative 99.8%

          \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + \color{blue}{x \cdot 2}\right) - 0.125 \cdot \frac{1}{x}} \]

       2. associate-*r/ 99.8%

          \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \color{blue}{\frac{0.125 \cdot 1}{x}}} \]

       3. metadata-eval 99.8%

          \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \frac{\color{blue}{0.125}}{x}} \]

   12. Simplified 99.8%

       \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\color{blue}{\left(0.5 + x \cdot 2\right) - \frac{0.125}{x}}} \]

   if 1.00000000000000004e-10 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.5%

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

      1. frac-sub 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. *-rgt-identity 99.5%

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

      4. +-commutative 99.5%

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

      5. sqrt-unprod 99.5%

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

      6. +-commutative 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. sqrt-prod 99.5%

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

      3. times-frac 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-flip 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. div-sub 99.9%

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

      2. *-inverses 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-10}:\\ \;\;\;\;{\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(1 - \frac{\sqrt{x}}{\sqrt{1 + x}}\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = pow((1.0 + x), -0.5) * (1.0 / (x + sqrt((x * (1.0 + x)))));
	}
	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)))) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = ((1.0d0 + x) ** (-0.5d0)) * (1.0d0 / (x + sqrt((x * (1.0d0 + x)))))
    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)))) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = Math.pow((1.0 + x), -0.5) * (1.0 / (x + Math.sqrt((x * (1.0 + x)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = math.pow((1.0 + x), -0.5) * (1.0 / (x + math.sqrt((x * (1.0 + x)))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = ((1.0 + x) ^ -0.5) * (1.0 / (x + sqrt((x * (1.0 + x)))));
	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], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 / N[(x + N[Sqrt[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 41.8%

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

      1. *-un-lft-identity 41.8%

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

      2. clear-num 41.8%

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

      3. associate-/r/ 41.8%

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

      4. prod-diff 41.8%

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

      5. *-un-lft-identity 41.8%

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

      6. fma-neg 41.8%

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

      7. *-un-lft-identity 41.8%

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

      8. pow1/2 41.8%

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

      9. pow-flip 31.3%

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

      10. metadata-eval 31.3%

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

      11. pow1/2 31.3%

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

      12. pow-flip 41.8%

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

      13. +-commutative 41.8%

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

      14. metadata-eval 41.8%

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

   3. Applied egg-rr 41.8%

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

      1. +-commutative 41.8%

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

      2. sub-neg 41.8%

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

      3. fma-udef 41.8%

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

      4. distribute-lft1-in 41.8%

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

      5. metadata-eval 41.8%

         \[\leadsto \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      6. mul0-lft 41.8%

         \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      7. +-commutative 41.8%

         \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

      8. associate-+r+ 41.8%

         \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

      9. sub-neg 41.8%

         \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      10. neg-sub0 41.8%

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

      11. +-commutative 41.8%

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

      12. sub-neg 41.8%

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

   5. Simplified 41.8%

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

      1. add-exp-log 5.5%

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

      2. log-pow 5.5%

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

      3. log1p-udef 5.5%

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

   7. Applied egg-rr 5.5%

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

   8. Taylor expanded in x around inf 74.7%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   9. Step-by-step derivation

      1. unpow-1 74.7%

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

      2. exp-to-pow 72.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\log \left({x}^{3}\right) \cdot -1}}} \]

      3. *-commutative 72.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \log \left({x}^{3}\right)}}} \]

      4. log-pow 72.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{-1 \cdot \color{blue}{\left(3 \cdot \log x\right)}}} \]

      5. associate-*r* 72.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

      6. metadata-eval 72.1%

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

      7. *-commutative 72.1%

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

      8. exp-to-pow 75.0%

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

      9. metadata-eval 75.0%

         \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

      10. pow-sqr 75.1%

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

      11. rem-sqrt-square 100.0%

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

      12. rem-square-sqrt 99.5%

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

      13. fabs-sqr 99.5%

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

      14. rem-square-sqrt 100.0%

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

   10. Simplified 100.0%

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

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

   1. Initial program 96.9%

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

      1. frac-sub 97.0%

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

      2. associate-/r* 97.0%

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

      3. *-un-lft-identity 97.0%

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

      4. *-rgt-identity 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

   3. Applied egg-rr 97.0%

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

      1. flip-- 97.8%

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

      2. div-inv 97.8%

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

      3. add-sqr-sqrt 98.7%

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

      4. +-commutative 98.7%

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

      5. add-sqr-sqrt 99.6%

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

      6. associate--l+ 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

      3. associate-+r- 99.6%

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

      4. +-commutative 99.6%

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

      5. associate--l+ 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-/r/ 99.6%

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

      4. inv-pow 99.6%

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

      5. sqrt-pow2 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-/l/ 99.6%

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

      9. +-inverses 99.6%

         \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1 + \color{blue}{0}}{\sqrt{x} \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]

      10. metadata-eval 99.6%

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

      11. distribute-lft-in 99.6%

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

      12. add-sqr-sqrt 99.6%

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

      13. sqrt-unprod 99.6%

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

      14. +-commutative 99.6%

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

   9. Applied egg-rr 99.6%

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

4. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = pow((1.0 + x), -0.5) / (x + sqrt((x * (1.0 + x))));
	}
	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)))) <= 0.0d0) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = ((1.0d0 + x) ** (-0.5d0)) / (x + sqrt((x * (1.0d0 + x))))
    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)))) <= 0.0) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = Math.pow((1.0 + x), -0.5) / (x + Math.sqrt((x * (1.0 + x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 0.0:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = math.pow((1.0 + x), -0.5) / (x + math.sqrt((x * (1.0 + x))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 0.0)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = ((1.0 + x) ^ -0.5) / (x + sqrt((x * (1.0 + x))));
	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], 0.0], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] / N[(x + N[Sqrt[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $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 41.8%

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

      1. *-un-lft-identity 41.8%

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

      2. clear-num 41.8%

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

      3. associate-/r/ 41.8%

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

      4. prod-diff 41.8%

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

      5. *-un-lft-identity 41.8%

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

      6. fma-neg 41.8%

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

      7. *-un-lft-identity 41.8%

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

      8. pow1/2 41.8%

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

      9. pow-flip 31.3%

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

      10. metadata-eval 31.3%

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

      11. pow1/2 31.3%

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

      12. pow-flip 41.8%

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

      13. +-commutative 41.8%

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

      14. metadata-eval 41.8%

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

   3. Applied egg-rr 41.8%

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

      1. +-commutative 41.8%

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

      2. sub-neg 41.8%

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

      3. fma-udef 41.8%

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

      4. distribute-lft1-in 41.8%

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

      5. metadata-eval 41.8%

         \[\leadsto \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      6. mul0-lft 41.8%

         \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      7. +-commutative 41.8%

         \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

      8. associate-+r+ 41.8%

         \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

      9. sub-neg 41.8%

         \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      10. neg-sub0 41.8%

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

      11. +-commutative 41.8%

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

      12. sub-neg 41.8%

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

   5. Simplified 41.8%

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

      1. add-exp-log 5.5%

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

      2. log-pow 5.5%

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

      3. log1p-udef 5.5%

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

   7. Applied egg-rr 5.5%

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

   8. Taylor expanded in x around inf 74.7%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   9. Step-by-step derivation

      1. unpow-1 74.7%

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

      2. exp-to-pow 72.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\log \left({x}^{3}\right) \cdot -1}}} \]

      3. *-commutative 72.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \log \left({x}^{3}\right)}}} \]

      4. log-pow 72.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{-1 \cdot \color{blue}{\left(3 \cdot \log x\right)}}} \]

      5. associate-*r* 72.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

      6. metadata-eval 72.1%

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

      7. *-commutative 72.1%

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

      8. exp-to-pow 75.0%

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

      9. metadata-eval 75.0%

         \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

      10. pow-sqr 75.1%

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

      11. rem-sqrt-square 100.0%

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

      12. rem-square-sqrt 99.5%

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

      13. fabs-sqr 99.5%

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

      14. rem-square-sqrt 100.0%

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

   10. Simplified 100.0%

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

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

   1. Initial program 96.9%

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

      1. frac-sub 97.0%

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

      2. associate-/r* 97.0%

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

      3. *-un-lft-identity 97.0%

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

      4. *-rgt-identity 97.0%

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

      5. +-commutative 97.0%

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

      6. +-commutative 97.0%

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

   3. Applied egg-rr 97.0%

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

      1. flip-- 97.8%

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

      2. div-inv 97.8%

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

      3. add-sqr-sqrt 98.7%

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

      4. +-commutative 98.7%

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

      5. add-sqr-sqrt 99.6%

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

      6. associate--l+ 99.6%

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

      7. +-commutative 99.6%

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

      8. +-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.6%

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

      2. *-rgt-identity 99.6%

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

      3. associate-+r- 99.6%

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

      4. +-commutative 99.6%

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

      5. associate--l+ 99.6%

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

   7. Simplified 99.6%

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

      1. expm1-log1p-u 92.6%

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

      2. expm1-udef 88.7%

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

   9. Applied egg-rr 88.7%

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

       1. expm1-def 92.6%

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

       2. expm1-log1p 99.6%

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

       3. +-commutative 99.6%

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

       4. +-commutative 99.6%

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

   11. Simplified 99.6%

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

4. Final simplification 99.8%

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

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 10^{-10}:\\ \;\;\;\;{\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sqrt (+ 1.0 x)))))
   (if (<= (+ (/ 1.0 (sqrt x)) t_0) 1e-10)
     (* (pow (+ 1.0 x) -0.5) (/ 1.0 (- (+ 0.5 (* x 2.0)) (/ 0.125 x))))
     (+ (pow x -0.5) t_0))))

C

double code(double x) {
	double t_0 = -1.0 / sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + t_0) <= 1e-10) {
		tmp = pow((1.0 + x), -0.5) * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)));
	} else {
		tmp = pow(x, -0.5) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) / sqrt((1.0d0 + x))
    if (((1.0d0 / sqrt(x)) + t_0) <= 1d-10) then
        tmp = ((1.0d0 + x) ** (-0.5d0)) * (1.0d0 / ((0.5d0 + (x * 2.0d0)) - (0.125d0 / x)))
    else
        tmp = (x ** (-0.5d0)) + t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = -1.0 / Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + t_0) <= 1e-10) {
		tmp = Math.pow((1.0 + x), -0.5) * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)));
	} else {
		tmp = Math.pow(x, -0.5) + t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = -1.0 / math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + t_0) <= 1e-10:
		tmp = math.pow((1.0 + x), -0.5) * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)))
	else:
		tmp = math.pow(x, -0.5) + t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(-1.0 / sqrt(Float64(1.0 + x)))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + t_0) <= 1e-10)
		tmp = Float64((Float64(1.0 + x) ^ -0.5) * Float64(1.0 / Float64(Float64(0.5 + Float64(x * 2.0)) - Float64(0.125 / x))));
	else
		tmp = Float64((x ^ -0.5) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = -1.0 / sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + t_0) <= 1e-10)
		tmp = ((1.0 + x) ^ -0.5) * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)));
	else
		tmp = (x ^ -0.5) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], 1e-10], N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 / N[(N[(0.5 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1.00000000000000004e-10

   1. Initial program 41.9%

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

      1. frac-sub 42.0%

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

      2. associate-/r* 42.0%

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

      3. *-un-lft-identity 42.0%

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

      4. *-rgt-identity 42.0%

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

      5. +-commutative 42.0%

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

      6. +-commutative 42.0%

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

   3. Applied egg-rr 42.0%

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

      1. flip-- 42.7%

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

      2. div-inv 42.7%

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

      3. add-sqr-sqrt 24.7%

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

      4. +-commutative 24.7%

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

      5. add-sqr-sqrt 44.4%

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

      6. associate--l+ 44.4%

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

      7. +-commutative 44.4%

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

      8. +-commutative 44.4%

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

   5. Applied egg-rr 44.4%

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

      1. associate-*r/ 44.4%

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

      2. *-rgt-identity 44.4%

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

      3. associate-+r- 44.4%

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

      4. +-commutative 44.4%

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

      5. associate--l+ 99.3%

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

   7. Simplified 99.3%

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

      1. clear-num 98.7%

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

      2. +-commutative 98.7%

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

      3. associate-/r/ 99.3%

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

      4. inv-pow 99.3%

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

      5. sqrt-pow2 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-/l/ 99.3%

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

      9. +-inverses 99.3%

         \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1 + \color{blue}{0}}{\sqrt{x} \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]

      10. metadata-eval 99.3%

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

      11. distribute-lft-in 99.3%

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

      12. add-sqr-sqrt 99.5%

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

      13. sqrt-unprod 82.9%

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

      14. +-commutative 82.9%

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

   9. Applied egg-rr 82.9%

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

   10. Taylor expanded in x around inf 99.8%

       \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\color{blue}{\left(0.5 + 2 \cdot x\right) - 0.125 \cdot \frac{1}{x}}} \]
   11. Step-by-step derivation

       1. *-commutative 99.8%

          \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + \color{blue}{x \cdot 2}\right) - 0.125 \cdot \frac{1}{x}} \]

       2. associate-*r/ 99.8%

          \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \color{blue}{\frac{0.125 \cdot 1}{x}}} \]

       3. metadata-eval 99.8%

          \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \frac{\color{blue}{0.125}}{x}} \]

   12. Simplified 99.8%

       \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\color{blue}{\left(0.5 + x \cdot 2\right) - \frac{0.125}{x}}} \]

   if 1.00000000000000004e-10 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.5%

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

      1. expm1-log1p-u 92.2%

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

      2. expm1-udef 92.1%

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

      3. pow1/2 92.1%

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

      4. pow-flip 92.1%

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

      5. metadata-eval 92.1%

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

   3. Applied egg-rr 92.1%

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

      1. expm1-def 92.2%

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

      2. expm1-log1p 99.8%

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

   5. Simplified 99.8%

      \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{\sqrt{x + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-10}:\\ \;\;\;\;{\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \frac{0.125}{x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{\sqrt{1 + x}}\\ \end{array} \]

Alternative 5: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	return (((1.0 + (x - x)) / (Math.sqrt(x) + t_0)) / Math.sqrt(x)) / t_0;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	return (((1.0 + (x - x)) / (math.sqrt(x) + t_0)) / math.sqrt(x)) / t_0

Julia

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

MATLAB

function tmp = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = (((1.0 + (x - x)) / (sqrt(x) + t_0)) / sqrt(x)) / t_0;
end

Wolfram

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

Derivation

1. Initial program 69.3%

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

   1. frac-sub 69.4%

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

   2. associate-/r* 69.4%

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

   3. *-un-lft-identity 69.4%

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

   4. *-rgt-identity 69.4%

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

   5. +-commutative 69.4%

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

   6. +-commutative 69.4%

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

3. Applied egg-rr 69.4%

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

   1. flip-- 69.8%

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

   2. div-inv 69.8%

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

   3. add-sqr-sqrt 60.4%

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

   4. +-commutative 60.4%

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

   5. add-sqr-sqrt 70.7%

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

   6. associate--l+ 70.7%

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

   7. +-commutative 70.7%

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

   8. +-commutative 70.7%

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

5. Applied egg-rr 70.7%

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

   1. associate-*r/ 70.7%

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

   2. *-rgt-identity 70.7%

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

   3. associate-+r- 70.7%

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

   4. +-commutative 70.7%

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

   5. associate--l+ 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 6: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(1 + x\right)}^{-0.5}\\ \mathbf{if}\;x \leq 4500:\\ \;\;\;\;{x}^{-0.5} - t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \frac{0.125}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (+ 1.0 x) -0.5)))
   (if (<= x 4500.0)
     (- (pow x -0.5) t_0)
     (* t_0 (/ 1.0 (- (+ 0.5 (* x 2.0)) (/ 0.125 x)))))))

C

double code(double x) {
	double t_0 = pow((1.0 + x), -0.5);
	double tmp;
	if (x <= 4500.0) {
		tmp = pow(x, -0.5) - t_0;
	} else {
		tmp = t_0 * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 + x) ** (-0.5d0)
    if (x <= 4500.0d0) then
        tmp = (x ** (-0.5d0)) - t_0
    else
        tmp = t_0 * (1.0d0 / ((0.5d0 + (x * 2.0d0)) - (0.125d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.pow((1.0 + x), -0.5);
	double tmp;
	if (x <= 4500.0) {
		tmp = Math.pow(x, -0.5) - t_0;
	} else {
		tmp = t_0 * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.pow((1.0 + x), -0.5)
	tmp = 0
	if x <= 4500.0:
		tmp = math.pow(x, -0.5) - t_0
	else:
		tmp = t_0 * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + x) ^ -0.5
	tmp = 0.0
	if (x <= 4500.0)
		tmp = Float64((x ^ -0.5) - t_0);
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(Float64(0.5 + Float64(x * 2.0)) - Float64(0.125 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (1.0 + x) ^ -0.5;
	tmp = 0.0;
	if (x <= 4500.0)
		tmp = (x ^ -0.5) - t_0;
	else
		tmp = t_0 * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[x, 4500.0], N[(N[Power[x, -0.5], $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(N[(0.5 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4500

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. +-commutative 99.8%

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

      2. sub-neg 99.8%

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

      3. fma-udef 99.8%

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

      4. distribute-lft1-in 99.8%

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

      5. metadata-eval 99.8%

         \[\leadsto \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      6. mul0-lft 99.8%

         \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      7. +-commutative 99.8%

         \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

      8. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

      9. sub-neg 99.8%

         \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      10. neg-sub0 99.8%

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

      11. +-commutative 99.8%

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

      12. sub-neg 99.8%

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

   5. Simplified 99.8%

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

   if 4500 < x

   1. Initial program 41.9%

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

      1. frac-sub 42.0%

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

      2. associate-/r* 42.0%

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

      3. *-un-lft-identity 42.0%

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

      4. *-rgt-identity 42.0%

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

      5. +-commutative 42.0%

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

      6. +-commutative 42.0%

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

   3. Applied egg-rr 42.0%

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

      1. flip-- 42.7%

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

      2. div-inv 42.7%

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

      3. add-sqr-sqrt 24.7%

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

      4. +-commutative 24.7%

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

      5. add-sqr-sqrt 44.4%

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

      6. associate--l+ 44.4%

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

      7. +-commutative 44.4%

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

      8. +-commutative 44.4%

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

   5. Applied egg-rr 44.4%

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

      1. associate-*r/ 44.4%

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

      2. *-rgt-identity 44.4%

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

      3. associate-+r- 44.4%

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

      4. +-commutative 44.4%

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

      5. associate--l+ 99.3%

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

   7. Simplified 99.3%

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

      1. clear-num 98.7%

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

      2. +-commutative 98.7%

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

      3. associate-/r/ 99.3%

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

      4. inv-pow 99.3%

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

      5. sqrt-pow2 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-/l/ 99.3%

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

      9. +-inverses 99.3%

         \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1 + \color{blue}{0}}{\sqrt{x} \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]

      10. metadata-eval 99.3%

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

      11. distribute-lft-in 99.3%

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

      12. add-sqr-sqrt 99.5%

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

      13. sqrt-unprod 82.9%

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

      14. +-commutative 82.9%

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

   9. Applied egg-rr 82.9%

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

   10. Taylor expanded in x around inf 99.8%

       \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\color{blue}{\left(0.5 + 2 \cdot x\right) - 0.125 \cdot \frac{1}{x}}} \]
   11. Step-by-step derivation

       1. *-commutative 99.8%

          \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + \color{blue}{x \cdot 2}\right) - 0.125 \cdot \frac{1}{x}} \]

       2. associate-*r/ 99.8%

          \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \color{blue}{\frac{0.125 \cdot 1}{x}}} \]

       3. metadata-eval 99.8%

          \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \frac{\color{blue}{0.125}}{x}} \]

   12. Simplified 99.8%

       \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\color{blue}{\left(0.5 + x \cdot 2\right) - \frac{0.125}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4500:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \frac{0.125}{x}}\\ \end{array} \]

Alternative 7: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \frac{0.125}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.56)
   (+ (pow x -0.5) (- -1.0 (* x (+ -0.5 (* x 0.375)))))
   (* (pow (+ 1.0 x) -0.5) (/ 1.0 (- (+ 0.5 (* x 2.0)) (/ 0.125 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.56) {
		tmp = pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = pow((1.0 + x), -0.5) * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.56d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) - (x * ((-0.5d0) + (x * 0.375d0))))
    else
        tmp = ((1.0d0 + x) ** (-0.5d0)) * (1.0d0 / ((0.5d0 + (x * 2.0d0)) - (0.125d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.56) {
		tmp = Math.pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = Math.pow((1.0 + x), -0.5) * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.56:
		tmp = math.pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))))
	else:
		tmp = math.pow((1.0 + x), -0.5) * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.56)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 - Float64(x * Float64(-0.5 + Float64(x * 0.375)))));
	else
		tmp = Float64((Float64(1.0 + x) ^ -0.5) * Float64(1.0 / Float64(Float64(0.5 + Float64(x * 2.0)) - Float64(0.125 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.56)
		tmp = (x ^ -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	else
		tmp = ((1.0 + x) ^ -0.5) * (1.0 / ((0.5 + (x * 2.0)) - (0.125 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.56], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 - N[(x * N[(-0.5 + N[(x * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 / N[(N[(0.5 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.56000000000000005

   1. Initial program 99.6%

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

      1. expm1-log1p-u 92.2%

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

      2. expm1-udef 92.2%

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

      3. pow1/2 92.2%

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

      4. pow-flip 92.2%

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

      5. metadata-eval 92.2%

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

   3. Applied egg-rr 92.2%

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

      1. expm1-def 92.2%

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

      2. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.8%

      \[\leadsto {x}^{-0.5} - \color{blue}{\left(1 + \left(-0.5 \cdot x + 0.375 \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto {x}^{-0.5} - \left(1 + \left(\color{blue}{x \cdot -0.5} + 0.375 \cdot {x}^{2}\right)\right) \]

      2. *-commutative 98.8%

         \[\leadsto {x}^{-0.5} - \left(1 + \left(x \cdot -0.5 + \color{blue}{{x}^{2} \cdot 0.375}\right)\right) \]

      3. unpow2 98.8%

         \[\leadsto {x}^{-0.5} - \left(1 + \left(x \cdot -0.5 + \color{blue}{\left(x \cdot x\right)} \cdot 0.375\right)\right) \]

      4. associate-*l* 98.8%

         \[\leadsto {x}^{-0.5} - \left(1 + \left(x \cdot -0.5 + \color{blue}{x \cdot \left(x \cdot 0.375\right)}\right)\right) \]

      5. distribute-lft-out 98.8%

         \[\leadsto {x}^{-0.5} - \left(1 + \color{blue}{x \cdot \left(-0.5 + x \cdot 0.375\right)}\right) \]

   8. Simplified 98.8%

      \[\leadsto {x}^{-0.5} - \color{blue}{\left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right)} \]

   if 0.56000000000000005 < x

   1. Initial program 42.6%

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

      1. frac-sub 42.8%

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

      2. associate-/r* 42.8%

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

      3. *-un-lft-identity 42.8%

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

      4. *-rgt-identity 42.8%

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

      5. +-commutative 42.8%

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

      6. +-commutative 42.8%

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

   3. Applied egg-rr 42.8%

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

      1. flip-- 43.6%

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

      2. div-inv 43.6%

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

      3. add-sqr-sqrt 25.8%

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

      4. +-commutative 25.8%

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

      5. add-sqr-sqrt 45.2%

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

      6. associate--l+ 45.2%

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

      7. +-commutative 45.2%

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

      8. +-commutative 45.2%

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

   5. Applied egg-rr 45.2%

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

      1. associate-*r/ 45.2%

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

      2. *-rgt-identity 45.2%

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

      3. associate-+r- 45.2%

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

      4. +-commutative 45.2%

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

      5. associate--l+ 99.3%

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

   7. Simplified 99.3%

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

      1. clear-num 98.7%

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

      2. +-commutative 98.7%

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

      3. associate-/r/ 99.3%

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

      4. inv-pow 99.3%

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

      5. sqrt-pow2 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-/l/ 99.3%

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

      9. +-inverses 99.3%

         \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1 + \color{blue}{0}}{\sqrt{x} \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]

      10. metadata-eval 99.3%

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

      11. distribute-lft-in 99.3%

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

      12. add-sqr-sqrt 99.5%

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

      13. sqrt-unprod 83.1%

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

      14. +-commutative 83.1%

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

   9. Applied egg-rr 83.1%

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

   10. Taylor expanded in x around inf 99.1%

       \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\color{blue}{\left(0.5 + 2 \cdot x\right) - 0.125 \cdot \frac{1}{x}}} \]
   11. Step-by-step derivation

       1. *-commutative 99.1%

          \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + \color{blue}{x \cdot 2}\right) - 0.125 \cdot \frac{1}{x}} \]

       2. associate-*r/ 99.1%

          \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \color{blue}{\frac{0.125 \cdot 1}{x}}} \]

       3. metadata-eval 99.1%

          \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \frac{\color{blue}{0.125}}{x}} \]

   12. Simplified 99.1%

       \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\color{blue}{\left(0.5 + x \cdot 2\right) - \frac{0.125}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{-0.5} \cdot \frac{1}{\left(0.5 + x \cdot 2\right) - \frac{0.125}{x}}\\ \end{array} \]

Alternative 8: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{-0.5} \cdot \frac{1}{x + \left(x + 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.48)
   (+ (pow x -0.5) (- -1.0 (* x -0.5)))
   (* (pow (+ 1.0 x) -0.5) (/ 1.0 (+ x (+ x 0.5))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.48) {
		tmp = pow(x, -0.5) + (-1.0 - (x * -0.5));
	} else {
		tmp = pow((1.0 + x), -0.5) * (1.0 / (x + (x + 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.48d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) - (x * (-0.5d0)))
    else
        tmp = ((1.0d0 + x) ** (-0.5d0)) * (1.0d0 / (x + (x + 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.48) {
		tmp = Math.pow(x, -0.5) + (-1.0 - (x * -0.5));
	} else {
		tmp = Math.pow((1.0 + x), -0.5) * (1.0 / (x + (x + 0.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.48:
		tmp = math.pow(x, -0.5) + (-1.0 - (x * -0.5))
	else:
		tmp = math.pow((1.0 + x), -0.5) * (1.0 / (x + (x + 0.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.48)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 - Float64(x * -0.5)));
	else
		tmp = Float64((Float64(1.0 + x) ^ -0.5) * Float64(1.0 / Float64(x + Float64(x + 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.48)
		tmp = (x ^ -0.5) + (-1.0 - (x * -0.5));
	else
		tmp = ((1.0 + x) ^ -0.5) * (1.0 / (x + (x + 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.48], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 / N[(x + N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.47999999999999998

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      6. mul0-lft 100.0%

         \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      7. +-commutative 100.0%

         \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

      8. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

      9. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   8. Simplified 98.7%

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

   if 0.47999999999999998 < x

   1. Initial program 42.6%

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

      1. frac-sub 42.8%

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

      2. associate-/r* 42.8%

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

      3. *-un-lft-identity 42.8%

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

      4. *-rgt-identity 42.8%

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

      5. +-commutative 42.8%

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

      6. +-commutative 42.8%

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

   3. Applied egg-rr 42.8%

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

      1. flip-- 43.6%

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

      2. div-inv 43.6%

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

      3. add-sqr-sqrt 25.8%

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

      4. +-commutative 25.8%

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

      5. add-sqr-sqrt 45.2%

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

      6. associate--l+ 45.2%

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

      7. +-commutative 45.2%

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

      8. +-commutative 45.2%

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

   5. Applied egg-rr 45.2%

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

      1. associate-*r/ 45.2%

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

      2. *-rgt-identity 45.2%

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

      3. associate-+r- 45.2%

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

      4. +-commutative 45.2%

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

      5. associate--l+ 99.3%

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

   7. Simplified 99.3%

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

      1. clear-num 98.7%

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

      2. +-commutative 98.7%

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

      3. associate-/r/ 99.3%

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

      4. inv-pow 99.3%

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

      5. sqrt-pow2 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-/l/ 99.3%

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

      9. +-inverses 99.3%

         \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1 + \color{blue}{0}}{\sqrt{x} \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]

      10. metadata-eval 99.3%

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

      11. distribute-lft-in 99.3%

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

      12. add-sqr-sqrt 99.5%

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

      13. sqrt-unprod 83.1%

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

      14. +-commutative 83.1%

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

   9. Applied egg-rr 83.1%

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

   10. Taylor expanded in x around inf 98.8%

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

       1. +-commutative 98.8%

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

   12. Simplified 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{-0.5} \cdot \frac{1}{x + \left(x + 0.5\right)}\\ \end{array} \]

Alternative 9: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{-0.5} \cdot \frac{1}{x + \left(x + 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.56)
   (+ (pow x -0.5) (- -1.0 (* x (+ -0.5 (* x 0.375)))))
   (* (pow (+ 1.0 x) -0.5) (/ 1.0 (+ x (+ x 0.5))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.56) {
		tmp = pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = pow((1.0 + x), -0.5) * (1.0 / (x + (x + 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.56d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) - (x * ((-0.5d0) + (x * 0.375d0))))
    else
        tmp = ((1.0d0 + x) ** (-0.5d0)) * (1.0d0 / (x + (x + 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.56) {
		tmp = Math.pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = Math.pow((1.0 + x), -0.5) * (1.0 / (x + (x + 0.5)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.56:
		tmp = math.pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))))
	else:
		tmp = math.pow((1.0 + x), -0.5) * (1.0 / (x + (x + 0.5)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.56)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 - Float64(x * Float64(-0.5 + Float64(x * 0.375)))));
	else
		tmp = Float64((Float64(1.0 + x) ^ -0.5) * Float64(1.0 / Float64(x + Float64(x + 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.56)
		tmp = (x ^ -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	else
		tmp = ((1.0 + x) ^ -0.5) * (1.0 / (x + (x + 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.56], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 - N[(x * N[(-0.5 + N[(x * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 / N[(x + N[(x + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.56000000000000005

   1. Initial program 99.6%

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

      1. expm1-log1p-u 92.2%

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

      2. expm1-udef 92.2%

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

      3. pow1/2 92.2%

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

      4. pow-flip 92.2%

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

      5. metadata-eval 92.2%

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

   3. Applied egg-rr 92.2%

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

      1. expm1-def 92.2%

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

      2. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.8%

      \[\leadsto {x}^{-0.5} - \color{blue}{\left(1 + \left(-0.5 \cdot x + 0.375 \cdot {x}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto {x}^{-0.5} - \left(1 + \left(\color{blue}{x \cdot -0.5} + 0.375 \cdot {x}^{2}\right)\right) \]

      2. *-commutative 98.8%

         \[\leadsto {x}^{-0.5} - \left(1 + \left(x \cdot -0.5 + \color{blue}{{x}^{2} \cdot 0.375}\right)\right) \]

      3. unpow2 98.8%

         \[\leadsto {x}^{-0.5} - \left(1 + \left(x \cdot -0.5 + \color{blue}{\left(x \cdot x\right)} \cdot 0.375\right)\right) \]

      4. associate-*l* 98.8%

         \[\leadsto {x}^{-0.5} - \left(1 + \left(x \cdot -0.5 + \color{blue}{x \cdot \left(x \cdot 0.375\right)}\right)\right) \]

      5. distribute-lft-out 98.8%

         \[\leadsto {x}^{-0.5} - \left(1 + \color{blue}{x \cdot \left(-0.5 + x \cdot 0.375\right)}\right) \]

   8. Simplified 98.8%

      \[\leadsto {x}^{-0.5} - \color{blue}{\left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right)} \]

   if 0.56000000000000005 < x

   1. Initial program 42.6%

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

      1. frac-sub 42.8%

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

      2. associate-/r* 42.8%

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

      3. *-un-lft-identity 42.8%

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

      4. *-rgt-identity 42.8%

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

      5. +-commutative 42.8%

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

      6. +-commutative 42.8%

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

   3. Applied egg-rr 42.8%

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

      1. flip-- 43.6%

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

      2. div-inv 43.6%

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

      3. add-sqr-sqrt 25.8%

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

      4. +-commutative 25.8%

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

      5. add-sqr-sqrt 45.2%

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

      6. associate--l+ 45.2%

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

      7. +-commutative 45.2%

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

      8. +-commutative 45.2%

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

   5. Applied egg-rr 45.2%

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

      1. associate-*r/ 45.2%

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

      2. *-rgt-identity 45.2%

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

      3. associate-+r- 45.2%

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

      4. +-commutative 45.2%

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

      5. associate--l+ 99.3%

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

   7. Simplified 99.3%

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

      1. clear-num 98.7%

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

      2. +-commutative 98.7%

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

      3. associate-/r/ 99.3%

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

      4. inv-pow 99.3%

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

      5. sqrt-pow2 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

      8. associate-/l/ 99.3%

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

      9. +-inverses 99.3%

         \[\leadsto {\left(1 + x\right)}^{-0.5} \cdot \frac{1 + \color{blue}{0}}{\sqrt{x} \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)} \]

      10. metadata-eval 99.3%

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

      11. distribute-lft-in 99.3%

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

      12. add-sqr-sqrt 99.5%

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

      13. sqrt-unprod 83.1%

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

      14. +-commutative 83.1%

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

   9. Applied egg-rr 83.1%

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

   10. Taylor expanded in x around inf 98.8%

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

       1. +-commutative 98.8%

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

   12. Simplified 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(1 + x\right)}^{-0.5} \cdot \frac{1}{x + \left(x + 0.5\right)}\\ \end{array} \]

Alternative 10: 98.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = pow(x, -0.5) + (-1.0 - (x * -0.5));
	} else {
		tmp = 0.5 * pow(x, -1.5);
	}
	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)) + ((-1.0d0) - (x * (-0.5d0)))
    else
        tmp = 0.5d0 * (x ** (-1.5d0))
    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) + (-1.0 - (x * -0.5));
	} else {
		tmp = 0.5 * Math.pow(x, -1.5);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      6. mul0-lft 100.0%

         \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      7. +-commutative 100.0%

         \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

      8. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

      9. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.1%

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

      1. *-commutative 98.1%

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

   8. Simplified 98.1%

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

   if 1 < x

   1. Initial program 42.2%

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

      1. *-un-lft-identity 42.2%

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

      2. clear-num 42.2%

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

      3. associate-/r/ 42.2%

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

      4. prod-diff 42.2%

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

      5. *-un-lft-identity 42.2%

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

      6. fma-neg 42.2%

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

      7. *-un-lft-identity 42.2%

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

      8. pow1/2 42.2%

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

      9. pow-flip 32.2%

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

      10. metadata-eval 32.2%

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

      11. pow1/2 32.2%

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

      12. pow-flip 42.2%

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

      13. +-commutative 42.2%

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

      14. metadata-eval 42.2%

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

   3. Applied egg-rr 42.2%

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

      1. +-commutative 42.2%

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

      2. sub-neg 42.2%

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

      3. fma-udef 42.2%

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

      4. distribute-lft1-in 42.2%

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

      5. metadata-eval 42.2%

         \[\leadsto \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      6. mul0-lft 42.2%

         \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      7. +-commutative 42.2%

         \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

      8. associate-+r+ 42.2%

         \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

      9. sub-neg 42.2%

         \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      10. neg-sub0 42.2%

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

      11. +-commutative 42.2%

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

      12. sub-neg 42.2%

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

   5. Simplified 42.2%

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

      1. add-exp-log 7.5%

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

      2. log-pow 7.6%

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

      3. log1p-udef 7.5%

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

   7. Applied egg-rr 7.5%

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

   8. Taylor expanded in x around inf 74.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   9. Step-by-step derivation

      1. unpow-1 74.5%

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

      2. exp-to-pow 72.0%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\log \left({x}^{3}\right) \cdot -1}}} \]

      3. *-commutative 72.0%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \log \left({x}^{3}\right)}}} \]

      4. log-pow 72.0%

         \[\leadsto 0.5 \cdot \sqrt{e^{-1 \cdot \color{blue}{\left(3 \cdot \log x\right)}}} \]

      5. associate-*r* 72.0%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

      6. metadata-eval 72.0%

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

      7. *-commutative 72.0%

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

      8. exp-to-pow 74.8%

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

      9. metadata-eval 74.8%

         \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

      10. pow-sqr 74.9%

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

      11. rem-sqrt-square 98.5%

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

      12. rem-square-sqrt 98.0%

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

      13. fabs-sqr 98.0%

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

      14. rem-square-sqrt 98.5%

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

   10. Simplified 98.5%

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

4. Final simplification 98.3%

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

Alternative 11: 96.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.6%

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

      1. expm1-log1p-u 92.2%

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

      2. expm1-udef 92.2%

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

      3. pow1/2 92.2%

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

      4. pow-flip 92.2%

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

      5. metadata-eval 92.2%

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

   3. Applied egg-rr 92.2%

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

      1. expm1-def 92.2%

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

      2. expm1-log1p 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 96.3%

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

      1. sqrt-div 96.2%

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

      2. metadata-eval 96.2%

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

      3. expm1-log1p-u 89.0%

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

      4. expm1-udef 89.0%

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

      5. inv-pow 89.0%

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

      6. sqrt-pow2 89.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1 \]

      7. metadata-eval 89.0%

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

   8. Applied egg-rr 89.0%

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

      1. expm1-def 89.0%

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

      2. expm1-log1p 96.5%

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

   10. Simplified 96.5%

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

   if 0.5 < x

   1. Initial program 42.6%

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

      1. *-un-lft-identity 42.6%

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

      2. clear-num 42.6%

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

      3. associate-/r/ 42.6%

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

      4. prod-diff 42.6%

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

      5. *-un-lft-identity 42.6%

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

      6. fma-neg 42.6%

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

      7. *-un-lft-identity 42.6%

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

      8. pow1/2 42.6%

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

      9. pow-flip 32.7%

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

      10. metadata-eval 32.7%

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

      11. pow1/2 32.7%

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

      12. pow-flip 42.6%

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

      13. +-commutative 42.6%

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

      14. metadata-eval 42.6%

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

   3. Applied egg-rr 42.6%

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

      1. +-commutative 42.6%

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

      2. sub-neg 42.6%

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

      3. fma-udef 42.6%

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

      4. distribute-lft1-in 42.6%

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

      5. metadata-eval 42.6%

         \[\leadsto \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      6. mul0-lft 42.6%

         \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      7. +-commutative 42.6%

         \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

      8. associate-+r+ 42.6%

         \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

      9. sub-neg 42.6%

         \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      10. neg-sub0 42.6%

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

      11. +-commutative 42.6%

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

      12. sub-neg 42.6%

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

   5. Simplified 42.6%

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

      1. add-exp-log 8.2%

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

      2. log-pow 8.3%

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

      3. log1p-udef 8.2%

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

   7. Applied egg-rr 8.2%

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

   8. Taylor expanded in x around inf 74.1%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   9. Step-by-step derivation

      1. unpow-1 74.1%

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

      2. exp-to-pow 71.6%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\log \left({x}^{3}\right) \cdot -1}}} \]

      3. *-commutative 71.6%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \log \left({x}^{3}\right)}}} \]

      4. log-pow 71.6%

         \[\leadsto 0.5 \cdot \sqrt{e^{-1 \cdot \color{blue}{\left(3 \cdot \log x\right)}}} \]

      5. associate-*r* 71.6%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

      6. metadata-eval 71.6%

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

      7. *-commutative 71.6%

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

      8. exp-to-pow 74.4%

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

      9. metadata-eval 74.4%

         \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

      10. pow-sqr 74.4%

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

      11. rem-sqrt-square 97.9%

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

      12. rem-square-sqrt 97.4%

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

      13. fabs-sqr 97.4%

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

      14. rem-square-sqrt 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 97.2%

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

Alternative 12: 98.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.68) (+ (pow x -0.5) -1.0) (* 0.5 (pow x -1.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 0.5 * pow(x, -1.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 0.5d0 * (x ** (-1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 0.5 * Math.pow(x, -1.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.68:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 0.5 * math.pow(x, -1.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(0.5 * (x ^ -1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 0.5 * (x ^ -1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.68], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

         \[\leadsto \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      6. mul0-lft 100.0%

         \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      7. +-commutative 100.0%

         \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

      8. associate-+r+ 100.0%

         \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

      9. sub-neg 100.0%

         \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 98.5%

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

   if 0.680000000000000049 < x

   1. Initial program 42.6%

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

      1. *-un-lft-identity 42.6%

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

      2. clear-num 42.6%

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

      3. associate-/r/ 42.6%

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

      4. prod-diff 42.6%

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

      5. *-un-lft-identity 42.6%

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

      6. fma-neg 42.6%

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

      7. *-un-lft-identity 42.6%

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

      8. pow1/2 42.6%

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

      9. pow-flip 32.7%

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

      10. metadata-eval 32.7%

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

      11. pow1/2 32.7%

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

      12. pow-flip 42.6%

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

      13. +-commutative 42.6%

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

      14. metadata-eval 42.6%

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

   3. Applied egg-rr 42.6%

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

      1. +-commutative 42.6%

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

      2. sub-neg 42.6%

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

      3. fma-udef 42.6%

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

      4. distribute-lft1-in 42.6%

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

      5. metadata-eval 42.6%

         \[\leadsto \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      6. mul0-lft 42.6%

         \[\leadsto \color{blue}{0} + \left({x}^{-0.5} + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) \]

      7. +-commutative 42.6%

         \[\leadsto 0 + \color{blue}{\left(\left(-{\left(1 + x\right)}^{-0.5}\right) + {x}^{-0.5}\right)} \]

      8. associate-+r+ 42.6%

         \[\leadsto \color{blue}{\left(0 + \left(-{\left(1 + x\right)}^{-0.5}\right)\right) + {x}^{-0.5}} \]

      9. sub-neg 42.6%

         \[\leadsto \color{blue}{\left(0 - {\left(1 + x\right)}^{-0.5}\right)} + {x}^{-0.5} \]

      10. neg-sub0 42.6%

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

      11. +-commutative 42.6%

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

      12. sub-neg 42.6%

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

   5. Simplified 42.6%

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

      1. add-exp-log 8.2%

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

      2. log-pow 8.3%

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

      3. log1p-udef 8.2%

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

   7. Applied egg-rr 8.2%

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

   8. Taylor expanded in x around inf 74.1%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   9. Step-by-step derivation

      1. unpow-1 74.1%

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

      2. exp-to-pow 71.6%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\log \left({x}^{3}\right) \cdot -1}}} \]

      3. *-commutative 71.6%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \log \left({x}^{3}\right)}}} \]

      4. log-pow 71.6%

         \[\leadsto 0.5 \cdot \sqrt{e^{-1 \cdot \color{blue}{\left(3 \cdot \log x\right)}}} \]

      5. associate-*r* 71.6%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

      6. metadata-eval 71.6%

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

      7. *-commutative 71.6%

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

      8. exp-to-pow 74.4%

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

      9. metadata-eval 74.4%

         \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

      10. pow-sqr 74.4%

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

      11. rem-sqrt-square 97.9%

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

      12. rem-square-sqrt 97.4%

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

      13. fabs-sqr 97.4%

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

      14. rem-square-sqrt 97.9%

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

   10. Simplified 97.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \end{array} \]

Alternative 13: 50.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 69.3%

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

   1. expm1-log1p-u 65.9%

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

   2. expm1-udef 46.2%

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

   3. pow1/2 46.2%

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

   4. pow-flip 46.2%

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

   5. metadata-eval 46.2%

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

3. Applied egg-rr 46.2%

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

   1. expm1-def 60.6%

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

   2. expm1-log1p 64.3%

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

5. Simplified 64.3%

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

6. Taylor expanded in x around inf 48.2%

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

   1. sqrt-div 48.1%

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

   2. metadata-eval 48.1%

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

   3. expm1-log1p-u 44.8%

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

   4. expm1-udef 63.2%

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

   5. inv-pow 63.2%

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

   6. sqrt-pow2 63.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1 \]

   7. metadata-eval 63.2%

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

8. Applied egg-rr 63.2%

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

   1. expm1-def 44.8%

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

   2. expm1-log1p 48.3%

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

10. Simplified 48.3%

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

11. Final simplification 48.3%

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

Alternative 14: 5.2% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot 0.1875 + 0.5 \cdot \frac{1}{x}\right) - 0.25 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (+ (* x 0.1875) (* 0.5 (/ 1.0 x))) 0.25))

C

double code(double x) {
	return ((x * 0.1875) + (0.5 * (1.0 / x))) - 0.25;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * 0.1875d0) + (0.5d0 * (1.0d0 / x))) - 0.25d0
end function

Java

public static double code(double x) {
	return ((x * 0.1875) + (0.5 * (1.0 / x))) - 0.25;
}

Python

def code(x):
	return ((x * 0.1875) + (0.5 * (1.0 / x))) - 0.25

Julia

function code(x)
	return Float64(Float64(Float64(x * 0.1875) + Float64(0.5 * Float64(1.0 / x))) - 0.25)
end

MATLAB

function tmp = code(x)
	tmp = ((x * 0.1875) + (0.5 * (1.0 / x))) - 0.25;
end

Wolfram

code[x_] := N[(N[(N[(x * 0.1875), $MachinePrecision] + N[(0.5 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.25), $MachinePrecision]

Derivation

1. Initial program 69.3%

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

   1. frac-sub 69.4%

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

   2. associate-/r* 69.4%

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

   3. *-un-lft-identity 69.4%

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

   4. *-rgt-identity 69.4%

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

   5. +-commutative 69.4%

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

   6. +-commutative 69.4%

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

3. Applied egg-rr 69.4%

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

   1. flip-- 69.8%

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

   2. div-inv 69.8%

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

   3. add-sqr-sqrt 60.4%

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

   4. +-commutative 60.4%

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

   5. add-sqr-sqrt 70.7%

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

   6. associate--l+ 70.7%

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

   7. +-commutative 70.7%

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

   8. +-commutative 70.7%

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

5. Applied egg-rr 70.7%

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

   1. associate-*r/ 70.7%

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

   2. *-rgt-identity 70.7%

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

   3. associate-+r- 70.7%

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

   4. +-commutative 70.7%

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

   5. associate--l+ 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around inf 55.1%

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

9. Taylor expanded in x around 0 5.1%

   \[\leadsto \color{blue}{\left(0.1875 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.25} \]

10. Final simplification 5.1%

    \[\leadsto \left(x \cdot 0.1875 + 0.5 \cdot \frac{1}{x}\right) - 0.25 \]

Alternative 15: 4.8% accurate, 41.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x} + -0.25 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (/ 0.5 x) -0.25))

C

double code(double x) {
	return (0.5 / x) + -0.25;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.5d0 / x) + (-0.25d0)
end function

Java

public static double code(double x) {
	return (0.5 / x) + -0.25;
}

Python

def code(x):
	return (0.5 / x) + -0.25

Julia

function code(x)
	return Float64(Float64(0.5 / x) + -0.25)
end

MATLAB

function tmp = code(x)
	tmp = (0.5 / x) + -0.25;
end

Wolfram

code[x_] := N[(N[(0.5 / x), $MachinePrecision] + -0.25), $MachinePrecision]

Derivation

1. Initial program 69.3%

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

   1. frac-sub 69.4%

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

   2. associate-/r* 69.4%

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

   3. *-un-lft-identity 69.4%

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

   4. *-rgt-identity 69.4%

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

   5. +-commutative 69.4%

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

   6. +-commutative 69.4%

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

3. Applied egg-rr 69.4%

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

   1. flip-- 69.8%

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

   2. div-inv 69.8%

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

   3. add-sqr-sqrt 60.4%

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

   4. +-commutative 60.4%

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

   5. add-sqr-sqrt 70.7%

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

   6. associate--l+ 70.7%

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

   7. +-commutative 70.7%

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

   8. +-commutative 70.7%

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

5. Applied egg-rr 70.7%

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

   1. associate-*r/ 70.7%

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

   2. *-rgt-identity 70.7%

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

   3. associate-+r- 70.7%

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

   4. +-commutative 70.7%

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

   5. associate--l+ 99.4%

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

7. Simplified 99.4%

   \[\leadsto \frac{\frac{\color{blue}{\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x + 1}}}}{\sqrt{x}}}{\sqrt{1 + x}} \]

8. Taylor expanded in x around inf 55.1%

   \[\leadsto \frac{\color{blue}{\frac{0.5}{x}}}{\sqrt{1 + x}} \]

9. Taylor expanded in x around 0 4.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x} - 0.25} \]
10. Step-by-step derivation

    1. sub-neg 4.7%

       \[\leadsto \color{blue}{0.5 \cdot \frac{1}{x} + \left(-0.25\right)} \]

    2. associate-*r/ 4.7%

       \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{x}} + \left(-0.25\right) \]

    3. metadata-eval 4.7%

       \[\leadsto \frac{\color{blue}{0.5}}{x} + \left(-0.25\right) \]

    4. metadata-eval 4.7%

       \[\leadsto \frac{0.5}{x} + \color{blue}{-0.25} \]

11. Simplified 4.7%

    \[\leadsto \color{blue}{\frac{0.5}{x} + -0.25} \]

12. Final simplification 4.7%

    \[\leadsto \frac{0.5}{x} + -0.25 \]

Alternative 16: 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 69.3%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

2. Taylor expanded in x around 0 47.4%

   \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{1} \]

3. Taylor expanded in x around inf 2.0%

   \[\leadsto \color{blue}{-1} \]

4. Final simplification 2.0%

   \[\leadsto -1 \]

Developer target: 99.0% 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 2023302 
(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)))))