2sqrt (example 3.1)

Percentage Accurate: 53.2% → 99.7%

Time: 10.3s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

   1. flip-- 50.3%

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

   2. div-inv 50.3%

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

   3. add-sqr-sqrt 50.6%

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

   4. add-sqr-sqrt 51.1%

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

   5. associate--l+ 51.1%

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

   6. reciprocal-define 47.3%

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

4. Applied egg-rr 47.3%

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

   1. reciprocal-undefine 51.1%

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

   2. associate-*r/ 51.1%

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

   3. *-rgt-identity 51.1%

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

   4. +-commutative 51.1%

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

   5. associate-+l- 99.7%

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

   6. +-inverses 99.7%

      \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

   7. metadata-eval 99.7%

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

   8. reciprocal-undefine 65.6%

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

   9. +-commutative 65.6%

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

6. Simplified 65.6%

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

   1. reciprocal-undefine 99.7%

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

   2. +-commutative 99.7%

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

8. Applied egg-rr 99.7%

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

   1. add-sqr-sqrt 99.7%

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

   2. hypot-1-def 99.7%

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 4e-6) (* 0.5 (pow x -0.5)) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 4e-6) {
		tmp = 0.5 * pow(x, -0.5);
	} else {
		tmp = 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)) - sqrt(x)
    if (t_0 <= 4d-6) then
        tmp = 0.5d0 * (x ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 4e-6) {
		tmp = 0.5 * Math.pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 4e-6:
		tmp = 0.5 * math.pow(x, -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 4e-6)
		tmp = Float64(0.5 * (x ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 4e-6)
		tmp = 0.5 * (x ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-6], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 3.99999999999999982e-6

   1. Initial program 4.7%

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

      1. flip-- 5.2%

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

      2. div-inv 5.2%

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

      3. add-sqr-sqrt 6.0%

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

      4. add-sqr-sqrt 6.7%

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

      5. associate--l+ 6.7%

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

      6. reciprocal-define 5.0%

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

   4. Applied egg-rr 5.0%

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

      1. reciprocal-undefine 6.7%

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

      2. associate-*r/ 6.7%

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

      3. *-rgt-identity 6.7%

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

      4. +-commutative 6.7%

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

      5. associate-+l- 99.5%

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

      6. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      7. metadata-eval 99.5%

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

      8. reciprocal-undefine 39.9%

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

      9. +-commutative 39.9%

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

   6. Simplified 39.9%

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

      1. reciprocal-undefine 99.5%

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

      2. +-commutative 99.5%

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

      3. flip3-+ 63.8%

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

      4. associate-/r/ 63.8%

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

      5. reciprocal-define 26.6%

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

      6. +-commutative 26.6%

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

      7. sqrt-pow2 26.6%

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

      8. metadata-eval 26.6%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      9. pow1/2 26.6%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\color{blue}{\left({\left(x + 1\right)}^{0.5}\right)}}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      10. +-commutative 26.6%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left({\color{blue}{\left(1 + x\right)}}^{0.5}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      11. pow-pow 26.6%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + \color{blue}{{\left(1 + x\right)}^{\left(0.5 \cdot 3\right)}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      12. metadata-eval 26.6%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{\color{blue}{1.5}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

   8. Applied egg-rr 19.2%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{1.5}\right)\right) \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]
   9. Step-by-step derivation

      1. reciprocal-undefine 47.8%

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

      2. associate-*l/ 47.8%

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

      3. *-lft-identity 47.8%

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

      4. associate-+l+ 47.8%

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

      5. associate--l+ 47.8%

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

      6. count-2 47.8%

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

      7. fma-udef 47.8%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{\color{blue}{x \cdot x + x}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      8. rem-square-sqrt 47.8%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{x \cdot x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      9. hypot-def 64.1%

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

   10. Simplified 64.1%

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

   11. Taylor expanded in x around inf 63.8%

       \[\leadsto \frac{1 + \color{blue}{x}}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

   12. Taylor expanded in x around inf 99.5%

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

       1. rem-exp-log 92.0%

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

       2. exp-neg 92.0%

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

       3. unpow1/2 92.0%

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

       4. exp-prod 92.0%

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

       5. distribute-lft-neg-out 92.0%

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

       6. distribute-rgt-neg-in 92.0%

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

       7. metadata-eval 92.0%

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

       8. exp-to-pow 99.7%

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

   14. Simplified 99.7%

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

   if 3.99999999999999982e-6 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.7%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 4 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

   1. flip-- 50.3%

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

   2. div-inv 50.3%

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

   3. add-sqr-sqrt 50.6%

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

   4. add-sqr-sqrt 51.1%

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

   5. associate--l+ 51.1%

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

   6. reciprocal-define 47.3%

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

4. Applied egg-rr 47.3%

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

   1. reciprocal-undefine 51.1%

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

   2. associate-*r/ 51.1%

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

   3. *-rgt-identity 51.1%

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

   4. +-commutative 51.1%

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

   5. associate-+l- 99.7%

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

   6. +-inverses 99.7%

      \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

   7. metadata-eval 99.7%

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

   8. reciprocal-undefine 65.6%

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

   9. +-commutative 65.6%

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

6. Simplified 65.6%

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

   1. reciprocal-undefine 99.7%

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

   2. +-commutative 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.22)
   (- (+ 1.0 (* x (+ 0.5 (* x -0.125)))) (sqrt x))
   (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.22) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - sqrt(x);
	} else {
		tmp = 0.5 * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.22d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * (-0.125d0))))) - sqrt(x)
    else
        tmp = 0.5d0 * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.22) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.22:
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - math.sqrt(x)
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.22)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * -0.125)))) - sqrt(x));
	else
		tmp = Float64(0.5 * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.22)
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - sqrt(x);
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.22], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.21999999999999997

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. unpow2 99.2%

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

      3. associate-*r* 99.2%

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

      4. distribute-rgt-out 99.2%

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

      5. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 1.21999999999999997 < x

   1. Initial program 6.6%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 7.8%

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

      4. add-sqr-sqrt 8.7%

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

      5. associate--l+ 8.7%

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

      6. reciprocal-define 5.7%

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

   4. Applied egg-rr 5.7%

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

      1. reciprocal-undefine 8.7%

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

      2. associate-*r/ 8.7%

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

      3. *-rgt-identity 8.7%

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

      4. +-commutative 8.7%

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

      5. associate-+l- 99.5%

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

      6. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      7. metadata-eval 99.5%

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

      8. reciprocal-undefine 39.9%

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

      9. +-commutative 39.9%

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

   6. Simplified 39.9%

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

      1. reciprocal-undefine 99.5%

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

      2. +-commutative 99.5%

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

      3. flip3-+ 64.5%

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

      4. associate-/r/ 64.5%

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

      5. reciprocal-define 26.9%

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

      6. +-commutative 26.9%

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

      7. sqrt-pow2 26.9%

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

      8. metadata-eval 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      9. pow1/2 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\color{blue}{\left({\left(x + 1\right)}^{0.5}\right)}}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      10. +-commutative 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left({\color{blue}{\left(1 + x\right)}}^{0.5}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      11. pow-pow 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + \color{blue}{{\left(1 + x\right)}^{\left(0.5 \cdot 3\right)}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      12. metadata-eval 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{\color{blue}{1.5}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

   8. Applied egg-rr 19.6%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{1.5}\right)\right) \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]
   9. Step-by-step derivation

      1. reciprocal-undefine 48.9%

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

      2. associate-*l/ 48.9%

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

      3. *-lft-identity 48.9%

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

      4. associate-+l+ 48.9%

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

      5. associate--l+ 48.9%

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

      6. count-2 48.9%

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

      7. fma-udef 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{\color{blue}{x \cdot x + x}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      8. rem-square-sqrt 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{x \cdot x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      9. hypot-def 64.9%

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

   10. Simplified 64.9%

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

   11. Taylor expanded in x around inf 63.1%

       \[\leadsto \frac{1 + \color{blue}{x}}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

   12. Taylor expanded in x around inf 98.1%

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

       1. rem-exp-log 90.7%

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

       2. exp-neg 90.7%

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

       3. unpow1/2 90.7%

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

       4. exp-prod 90.7%

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

       5. distribute-lft-neg-out 90.7%

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

       6. distribute-rgt-neg-in 90.7%

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

       7. metadata-eval 90.7%

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

       8. exp-to-pow 98.3%

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

   14. Simplified 98.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 98.5%

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

      1. associate--l+ 98.5%

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

      2. +-commutative 98.5%

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

      3. *-commutative 98.5%

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

   5. Applied egg-rr 98.5%

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

   if 1 < x

   1. Initial program 6.6%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 7.8%

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

      4. add-sqr-sqrt 8.7%

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

      5. associate--l+ 8.7%

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

      6. reciprocal-define 5.7%

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

   4. Applied egg-rr 5.7%

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

      1. reciprocal-undefine 8.7%

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

      2. associate-*r/ 8.7%

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

      3. *-rgt-identity 8.7%

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

      4. +-commutative 8.7%

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

      5. associate-+l- 99.5%

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

      6. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      7. metadata-eval 99.5%

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

      8. reciprocal-undefine 39.9%

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

      9. +-commutative 39.9%

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

   6. Simplified 39.9%

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

      1. reciprocal-undefine 99.5%

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

      2. +-commutative 99.5%

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

      3. flip3-+ 64.5%

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

      4. associate-/r/ 64.5%

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

      5. reciprocal-define 26.9%

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

      6. +-commutative 26.9%

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

      7. sqrt-pow2 26.9%

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

      8. metadata-eval 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      9. pow1/2 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\color{blue}{\left({\left(x + 1\right)}^{0.5}\right)}}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      10. +-commutative 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left({\color{blue}{\left(1 + x\right)}}^{0.5}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      11. pow-pow 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + \color{blue}{{\left(1 + x\right)}^{\left(0.5 \cdot 3\right)}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      12. metadata-eval 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{\color{blue}{1.5}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

   8. Applied egg-rr 19.6%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{1.5}\right)\right) \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]
   9. Step-by-step derivation

      1. reciprocal-undefine 48.9%

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

      2. associate-*l/ 48.9%

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

      3. *-lft-identity 48.9%

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

      4. associate-+l+ 48.9%

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

      5. associate--l+ 48.9%

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

      6. count-2 48.9%

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

      7. fma-udef 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{\color{blue}{x \cdot x + x}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      8. rem-square-sqrt 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{x \cdot x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      9. hypot-def 64.9%

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

   10. Simplified 64.9%

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

   11. Taylor expanded in x around inf 63.1%

       \[\leadsto \frac{1 + \color{blue}{x}}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

   12. Taylor expanded in x around inf 98.1%

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

       1. rem-exp-log 90.7%

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

       2. exp-neg 90.7%

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

       3. unpow1/2 90.7%

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

       4. exp-prod 90.7%

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

       5. distribute-lft-neg-out 90.7%

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

       6. distribute-rgt-neg-in 90.7%

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

       7. metadata-eval 90.7%

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

       8. exp-to-pow 98.3%

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

   14. Simplified 98.3%

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

4. Final simplification 98.4%

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

Alternative 6: 96.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;\frac{1}{1 + {x}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.38) (/ 1.0 (+ 1.0 (pow x 1.5))) (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.38) {
		tmp = 1.0 / (1.0 + pow(x, 1.5));
	} else {
		tmp = 0.5 * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.38d0) then
        tmp = 1.0d0 / (1.0d0 + (x ** 1.5d0))
    else
        tmp = 0.5d0 * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.38) {
		tmp = 1.0 / (1.0 + Math.pow(x, 1.5));
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.38:
		tmp = 1.0 / (1.0 + math.pow(x, 1.5))
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.38)
		tmp = Float64(1.0 / Float64(1.0 + (x ^ 1.5)));
	else
		tmp = Float64(0.5 * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.38)
		tmp = 1.0 / (1.0 + (x ^ 1.5));
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.38], N[(1.0 / N[(1.0 + N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.38

   1. Initial program 99.9%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--l+ 99.9%

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

      6. reciprocal-define 95.1%

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

   4. Applied egg-rr 95.1%

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

      1. reciprocal-undefine 99.9%

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

      2. associate-*r/ 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l- 99.9%

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

      6. +-inverses 99.9%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      7. metadata-eval 99.9%

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

      8. reciprocal-undefine 95.1%

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

      9. +-commutative 95.1%

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

   6. Simplified 95.1%

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

      1. reciprocal-undefine 99.9%

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

      2. +-commutative 99.9%

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

      3. flip3-+ 99.9%

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

      4. associate-/r/ 99.9%

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

      5. reciprocal-define 96.8%

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

      6. +-commutative 96.8%

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

      7. sqrt-pow2 96.8%

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

      8. metadata-eval 96.8%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      9. pow1/2 96.8%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\color{blue}{\left({\left(x + 1\right)}^{0.5}\right)}}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      10. +-commutative 96.8%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left({\color{blue}{\left(1 + x\right)}}^{0.5}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      11. pow-pow 96.8%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + \color{blue}{{\left(1 + x\right)}^{\left(0.5 \cdot 3\right)}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      12. metadata-eval 96.8%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{\color{blue}{1.5}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

   8. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{1.5}\right)\right) \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]
   9. Step-by-step derivation

      1. reciprocal-undefine 99.9%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-+l+ 99.9%

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

      5. associate--l+ 99.9%

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

      6. count-2 99.9%

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

      7. fma-udef 99.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{\color{blue}{x \cdot x + x}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      8. rem-square-sqrt 99.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{x \cdot x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      9. hypot-def 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in x around 0 94.6%

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

   if 0.38 < x

   1. Initial program 6.6%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 7.8%

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

      4. add-sqr-sqrt 8.7%

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

      5. associate--l+ 8.7%

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

      6. reciprocal-define 5.7%

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

   4. Applied egg-rr 5.7%

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

      1. reciprocal-undefine 8.7%

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

      2. associate-*r/ 8.7%

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

      3. *-rgt-identity 8.7%

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

      4. +-commutative 8.7%

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

      5. associate-+l- 99.5%

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

      6. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      7. metadata-eval 99.5%

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

      8. reciprocal-undefine 39.9%

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

      9. +-commutative 39.9%

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

   6. Simplified 39.9%

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

      1. reciprocal-undefine 99.5%

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

      2. +-commutative 99.5%

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

      3. flip3-+ 64.5%

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

      4. associate-/r/ 64.5%

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

      5. reciprocal-define 26.9%

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

      6. +-commutative 26.9%

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

      7. sqrt-pow2 26.9%

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

      8. metadata-eval 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      9. pow1/2 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\color{blue}{\left({\left(x + 1\right)}^{0.5}\right)}}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      10. +-commutative 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left({\color{blue}{\left(1 + x\right)}}^{0.5}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      11. pow-pow 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + \color{blue}{{\left(1 + x\right)}^{\left(0.5 \cdot 3\right)}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      12. metadata-eval 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{\color{blue}{1.5}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

   8. Applied egg-rr 19.6%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{1.5}\right)\right) \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]
   9. Step-by-step derivation

      1. reciprocal-undefine 48.9%

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

      2. associate-*l/ 48.9%

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

      3. *-lft-identity 48.9%

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

      4. associate-+l+ 48.9%

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

      5. associate--l+ 48.9%

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

      6. count-2 48.9%

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

      7. fma-udef 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{\color{blue}{x \cdot x + x}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      8. rem-square-sqrt 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{x \cdot x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      9. hypot-def 64.9%

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

   10. Simplified 64.9%

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

   11. Taylor expanded in x around inf 63.1%

       \[\leadsto \frac{1 + \color{blue}{x}}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

   12. Taylor expanded in x around inf 98.1%

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

       1. rem-exp-log 90.7%

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

       2. exp-neg 90.7%

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

       3. unpow1/2 90.7%

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

       4. exp-prod 90.7%

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

       5. distribute-lft-neg-out 90.7%

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

       6. distribute-rgt-neg-in 90.7%

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

       7. metadata-eval 90.7%

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

       8. exp-to-pow 98.3%

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

   14. Simplified 98.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;\frac{1}{1 + {x}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(1 + {x}^{1.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.38) (reciprocal (+ 1.0 (pow x 1.5))) (* 0.5 (pow x -0.5))))

Derivation

1. Split input into 2 regimes

2. if x < 0.38

   1. Initial program 99.9%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--l+ 99.9%

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

      6. reciprocal-define 95.1%

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

   4. Applied egg-rr 95.1%

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

      1. reciprocal-undefine 99.9%

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

      2. associate-*r/ 99.9%

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

      3. *-rgt-identity 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+l- 99.9%

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

      6. +-inverses 99.9%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      7. metadata-eval 99.9%

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

      8. reciprocal-undefine 95.1%

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

      9. +-commutative 95.1%

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

   6. Simplified 95.1%

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

      1. reciprocal-undefine 99.9%

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

      2. +-commutative 99.9%

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

      3. flip3-+ 99.9%

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

      4. associate-/r/ 99.9%

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

      5. reciprocal-define 96.8%

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

      6. +-commutative 96.8%

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

      7. sqrt-pow2 96.8%

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

      8. metadata-eval 96.8%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      9. pow1/2 96.8%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\color{blue}{\left({\left(x + 1\right)}^{0.5}\right)}}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      10. +-commutative 96.8%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left({\color{blue}{\left(1 + x\right)}}^{0.5}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      11. pow-pow 96.8%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + \color{blue}{{\left(1 + x\right)}^{\left(0.5 \cdot 3\right)}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      12. metadata-eval 96.8%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{\color{blue}{1.5}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

   8. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{1.5}\right)\right) \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]
   9. Step-by-step derivation

      1. reciprocal-undefine 99.9%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

      4. associate-+l+ 99.9%

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

      5. associate--l+ 99.9%

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

      6. count-2 99.9%

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

      7. fma-udef 99.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{\color{blue}{x \cdot x + x}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      8. rem-square-sqrt 99.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{x \cdot x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      9. hypot-def 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in x around 0 94.6%

       \[\leadsto \color{blue}{\frac{1}{1 + {x}^{1.5}}} \]
   12. Step-by-step derivation

       1. reciprocal-define 94.6%

          \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {x}^{1.5}\right)\right)} \]

   13. Simplified 94.6%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(1 + {x}^{1.5}\right)\right)} \]

   if 0.38 < x

   1. Initial program 6.6%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 7.8%

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

      4. add-sqr-sqrt 8.7%

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

      5. associate--l+ 8.7%

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

      6. reciprocal-define 5.7%

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

   4. Applied egg-rr 5.7%

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

      1. reciprocal-undefine 8.7%

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

      2. associate-*r/ 8.7%

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

      3. *-rgt-identity 8.7%

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

      4. +-commutative 8.7%

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

      5. associate-+l- 99.5%

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

      6. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      7. metadata-eval 99.5%

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

      8. reciprocal-undefine 39.9%

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

      9. +-commutative 39.9%

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

   6. Simplified 39.9%

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

      1. reciprocal-undefine 99.5%

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

      2. +-commutative 99.5%

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

      3. flip3-+ 64.5%

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

      4. associate-/r/ 64.5%

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

      5. reciprocal-define 26.9%

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

      6. +-commutative 26.9%

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

      7. sqrt-pow2 26.9%

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

      8. metadata-eval 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      9. pow1/2 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\color{blue}{\left({\left(x + 1\right)}^{0.5}\right)}}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      10. +-commutative 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left({\color{blue}{\left(1 + x\right)}}^{0.5}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      11. pow-pow 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + \color{blue}{{\left(1 + x\right)}^{\left(0.5 \cdot 3\right)}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      12. metadata-eval 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{\color{blue}{1.5}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

   8. Applied egg-rr 19.6%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{1.5}\right)\right) \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]
   9. Step-by-step derivation

      1. reciprocal-undefine 48.9%

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

      2. associate-*l/ 48.9%

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

      3. *-lft-identity 48.9%

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

      4. associate-+l+ 48.9%

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

      5. associate--l+ 48.9%

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

      6. count-2 48.9%

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

      7. fma-udef 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{\color{blue}{x \cdot x + x}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      8. rem-square-sqrt 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{x \cdot x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      9. hypot-def 64.9%

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

   10. Simplified 64.9%

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

   11. Taylor expanded in x around inf 63.1%

       \[\leadsto \frac{1 + \color{blue}{x}}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

   12. Taylor expanded in x around inf 98.1%

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

       1. rem-exp-log 90.7%

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

       2. exp-neg 90.7%

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

       3. unpow1/2 90.7%

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

       4. exp-prod 90.7%

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

       5. distribute-lft-neg-out 90.7%

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

       6. distribute-rgt-neg-in 90.7%

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

       7. metadata-eval 90.7%

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

       8. exp-to-pow 98.3%

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

   14. Simplified 98.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.38:\\ \;\;\;\;\mathsf{reciprocal}\left(\left(1 + {x}^{1.5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.25) 1.0 (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.25d0) then
        tmp = 1.0d0
    else
        tmp = 0.5d0 * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.25:
		tmp = 1.0
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = Float64(0.5 * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.25], 1.0, N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 94.6%

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

   if 0.25 < x

   1. Initial program 6.6%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 7.8%

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

      4. add-sqr-sqrt 8.7%

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

      5. associate--l+ 8.7%

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

      6. reciprocal-define 5.7%

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

   4. Applied egg-rr 5.7%

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

      1. reciprocal-undefine 8.7%

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

      2. associate-*r/ 8.7%

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

      3. *-rgt-identity 8.7%

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

      4. +-commutative 8.7%

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

      5. associate-+l- 99.5%

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

      6. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      7. metadata-eval 99.5%

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

      8. reciprocal-undefine 39.9%

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

      9. +-commutative 39.9%

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

   6. Simplified 39.9%

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

      1. reciprocal-undefine 99.5%

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

      2. +-commutative 99.5%

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

      3. flip3-+ 64.5%

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

      4. associate-/r/ 64.5%

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

      5. reciprocal-define 26.9%

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

      6. +-commutative 26.9%

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

      7. sqrt-pow2 26.9%

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

      8. metadata-eval 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      9. pow1/2 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\color{blue}{\left({\left(x + 1\right)}^{0.5}\right)}}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      10. +-commutative 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left({\color{blue}{\left(1 + x\right)}}^{0.5}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      11. pow-pow 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + \color{blue}{{\left(1 + x\right)}^{\left(0.5 \cdot 3\right)}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      12. metadata-eval 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{\color{blue}{1.5}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

   8. Applied egg-rr 19.6%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{1.5}\right)\right) \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]
   9. Step-by-step derivation

      1. reciprocal-undefine 48.9%

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

      2. associate-*l/ 48.9%

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

      3. *-lft-identity 48.9%

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

      4. associate-+l+ 48.9%

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

      5. associate--l+ 48.9%

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

      6. count-2 48.9%

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

      7. fma-udef 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{\color{blue}{x \cdot x + x}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      8. rem-square-sqrt 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{x \cdot x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      9. hypot-def 64.9%

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

   10. Simplified 64.9%

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

   11. Taylor expanded in x around inf 63.1%

       \[\leadsto \frac{1 + \color{blue}{x}}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

   12. Taylor expanded in x around inf 98.1%

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

       1. rem-exp-log 90.7%

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

       2. exp-neg 90.7%

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

       3. unpow1/2 90.7%

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

       4. exp-prod 90.7%

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

       5. distribute-lft-neg-out 90.7%

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

       6. distribute-rgt-neg-in 90.7%

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

       7. metadata-eval 90.7%

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

       8. exp-to-pow 98.3%

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

   14. Simplified 98.3%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.25) 1.0 (* 0.5 (reciprocal (sqrt x)))))

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 94.6%

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

   if 0.25 < x

   1. Initial program 6.6%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 7.8%

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

      4. add-sqr-sqrt 8.7%

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

      5. associate--l+ 8.7%

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

      6. reciprocal-define 5.7%

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

   4. Applied egg-rr 5.7%

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

      1. reciprocal-undefine 8.7%

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

      2. associate-*r/ 8.7%

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

      3. *-rgt-identity 8.7%

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

      4. +-commutative 8.7%

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

      5. associate-+l- 99.5%

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

      6. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      7. metadata-eval 99.5%

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

      8. reciprocal-undefine 39.9%

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

      9. +-commutative 39.9%

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

   6. Simplified 39.9%

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

      1. reciprocal-undefine 99.5%

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

      2. +-commutative 99.5%

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

      3. flip3-+ 64.5%

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

      4. associate-/r/ 64.5%

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

      5. reciprocal-define 26.9%

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

      6. +-commutative 26.9%

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

      7. sqrt-pow2 26.9%

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

      8. metadata-eval 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      9. pow1/2 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\color{blue}{\left({\left(x + 1\right)}^{0.5}\right)}}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      10. +-commutative 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left({\color{blue}{\left(1 + x\right)}}^{0.5}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      11. pow-pow 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + \color{blue}{{\left(1 + x\right)}^{\left(0.5 \cdot 3\right)}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      12. metadata-eval 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{\color{blue}{1.5}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

   8. Applied egg-rr 19.6%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{1.5}\right)\right) \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]
   9. Step-by-step derivation

      1. reciprocal-undefine 48.9%

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

      2. associate-*l/ 48.9%

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

      3. *-lft-identity 48.9%

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

      4. associate-+l+ 48.9%

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

      5. associate--l+ 48.9%

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

      6. count-2 48.9%

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

      7. fma-udef 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{\color{blue}{x \cdot x + x}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      8. rem-square-sqrt 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{x \cdot x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      9. hypot-def 64.9%

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

   10. Simplified 64.9%

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

   11. Taylor expanded in x around inf 98.1%

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

       1. sqrt-div 97.9%

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

       2. metadata-eval 97.9%

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

       3. reciprocal-define 39.7%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right)} \]

   13. Applied egg-rr 39.7%

       \[\leadsto 0.5 \cdot \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{reciprocal}\left(x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.25) 1.0 (* 0.5 (sqrt (reciprocal x)))))

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 94.6%

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

   if 0.25 < x

   1. Initial program 6.6%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 7.8%

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

      4. add-sqr-sqrt 8.7%

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

      5. associate--l+ 8.7%

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

      6. reciprocal-define 5.7%

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

   4. Applied egg-rr 5.7%

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

      1. reciprocal-undefine 8.7%

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

      2. associate-*r/ 8.7%

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

      3. *-rgt-identity 8.7%

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

      4. +-commutative 8.7%

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

      5. associate-+l- 99.5%

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

      6. +-inverses 99.5%

         \[\leadsto \frac{1 - \color{blue}{0}}{\sqrt{x + 1} + \sqrt{x}} \]

      7. metadata-eval 99.5%

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

      8. reciprocal-undefine 39.9%

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

      9. +-commutative 39.9%

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

   6. Simplified 39.9%

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

      1. reciprocal-undefine 99.5%

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

      2. +-commutative 99.5%

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

      3. flip3-+ 64.5%

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

      4. associate-/r/ 64.5%

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

      5. reciprocal-define 26.9%

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

      6. +-commutative 26.9%

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

      7. sqrt-pow2 26.9%

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

      8. metadata-eval 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{\color{blue}{1.5}} + {\left(\sqrt{x + 1}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      9. pow1/2 26.9%

         \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\color{blue}{\left({\left(x + 1\right)}^{0.5}\right)}}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      10. +-commutative 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left({\color{blue}{\left(1 + x\right)}}^{0.5}\right)}^{3}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      11. pow-pow 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + \color{blue}{{\left(1 + x\right)}^{\left(0.5 \cdot 3\right)}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

      12. metadata-eval 26.9%

          \[\leadsto \mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{\color{blue}{1.5}}\right)\right) \cdot \left(\sqrt{x + 1} \cdot \sqrt{x + 1} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{x + 1} \cdot \sqrt{x}\right)\right) \]

   8. Applied egg-rr 19.6%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left({x}^{1.5} + {\left(1 + x\right)}^{1.5}\right)\right) \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)} \]
   9. Step-by-step derivation

      1. reciprocal-undefine 48.9%

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

      2. associate-*l/ 48.9%

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

      3. *-lft-identity 48.9%

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

      4. associate-+l+ 48.9%

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

      5. associate--l+ 48.9%

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

      6. count-2 48.9%

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

      7. fma-udef 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{\color{blue}{x \cdot x + x}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      8. rem-square-sqrt 48.9%

         \[\leadsto \frac{1 + \left(2 \cdot x - \sqrt{x \cdot x + \color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}{{x}^{1.5} + {\left(1 + x\right)}^{1.5}} \]

      9. hypot-def 64.9%

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

   10. Simplified 64.9%

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

   11. Taylor expanded in x around inf 98.1%

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

       1. reciprocal-define 40.8%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\mathsf{reciprocal}\left(x\right)}} \]

   13. Simplified 40.8%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\mathsf{reciprocal}\left(x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{reciprocal}\left(x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.1% accurate, 205.0× 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 50.0%

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

3. Taylor expanded in x around 0 47.7%

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

4. Final simplification 47.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 66000000:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x + 1} + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 66000000.0)
   (- (sqrt (+ 1.0 x)) (sqrt x))
   (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x)))))

C

double code(double x) {
	double tmp;
	if (x <= 66000000.0) {
		tmp = sqrt((1.0 + x)) - sqrt(x);
	} else {
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 66000000.0d0) then
        tmp = sqrt((1.0d0 + x)) - sqrt(x)
    else
        tmp = 1.0d0 / (sqrt((x + 1.0d0)) + sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 66000000.0) {
		tmp = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	} else {
		tmp = 1.0 / (Math.sqrt((x + 1.0)) + Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 66000000.0:
		tmp = math.sqrt((1.0 + x)) - math.sqrt(x)
	else:
		tmp = 1.0 / (math.sqrt((x + 1.0)) + math.sqrt(x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 66000000.0)
		tmp = Float64(sqrt(Float64(1.0 + x)) - sqrt(x));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(x + 1.0)) + sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 66000000.0)
		tmp = sqrt((1.0 + x)) - sqrt(x);
	else
		tmp = 1.0 / (sqrt((x + 1.0)) + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 66000000.0], N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024024 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

  :herbie-target
  (if (<= x 66000000.0) (- (sqrt (+ 1.0 x)) (sqrt x)) (/ 1.0 (+ (sqrt (+ x 1.0)) (sqrt x))))

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