2sqrt (example 3.1)

Percentage Accurate: 54.1% → 99.7%

Time: 7.8s

Alternatives: 12

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: 54.1% 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, 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 54.1%

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

   1. flip-- 54.5%

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

   2. rem-square-sqrt 54.6%

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

   3. div-sub 54.0%

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

   4. sub-neg 54.0%

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

   5. rem-square-sqrt 54.1%

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

3. Applied egg-rr 54.1%

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

   1. unsub-neg 54.1%

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

   2. div-sub 55.6%

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

   3. +-commutative 55.6%

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

   4. associate--l+ 99.7%

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

   5. +-inverses 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 5e-5) (* 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 <= 5e-5) {
		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 <= 5d-5) 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 <= 5e-5) {
		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 <= 5e-5:
		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 <= 5e-5)
		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 <= 5e-5)
		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, 5e-5], 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)) < 5.00000000000000024e-5

   1. Initial program 5.5%

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

      1. flip-- 6.0%

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

      2. flip3-+ 6.0%

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

      3. associate-/r/ 6.0%

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

      4. *-commutative 6.0%

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

   3. Applied egg-rr 5.3%

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

      1. +-commutative 5.3%

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

      2. associate-+l- 41.5%

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

      3. +-inverses 41.5%

         \[\leadsto \left(x + \left(1 + \left(x - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)\right)\right) \cdot \frac{1 - \color{blue}{0}}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \]

      4. metadata-eval 41.5%

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

      5. associate-*r/ 41.5%

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

      6. associate-/l* 41.5%

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

   5. Simplified 65.6%

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

   6. Taylor expanded in x around inf 64.6%

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

   7. Taylor expanded in x around inf 98.9%

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

      1. unpow-1 98.9%

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

      2. sqr-pow 99.1%

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

      3. metadata-eval 99.1%

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

      4. metadata-eval 99.1%

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

      5. rem-sqrt-square 99.1%

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

      6. rem-square-sqrt 98.3%

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

      7. fabs-sqr 98.3%

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

      8. rem-square-sqrt 99.1%

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

   9. Simplified 99.1%

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

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.0%

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

4. Final simplification 99.1%

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

Alternative 3: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25)
   (- (+ 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 <= 1.25) {
		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 <= 1.25d0) 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 <= 1.25) {
		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 <= 1.25:
		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 <= 1.25)
		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 <= 1.25)
		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, 1.25], 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 < 1.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. *-commutative 98.6%

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

      3. *-commutative 98.6%

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

      4. unpow2 98.6%

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

      5. associate-*l* 98.6%

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

      6. distribute-lft-out 98.6%

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

   4. Simplified 98.6%

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

   if 1.25 < x

   1. Initial program 7.5%

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

      1. flip-- 8.3%

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

      2. flip3-+ 8.3%

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

      3. associate-/r/ 8.3%

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

      4. *-commutative 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. +-commutative 8.3%

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

      2. associate-+l- 43.3%

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

      3. +-inverses 43.3%

         \[\leadsto \left(x + \left(1 + \left(x - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)\right)\right) \cdot \frac{1 - \color{blue}{0}}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \]

      4. metadata-eval 43.3%

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

      5. associate-*r/ 43.4%

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

      6. associate-/l* 43.4%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in x around inf 64.2%

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

   7. Taylor expanded in x around inf 97.5%

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

      1. unpow-1 97.5%

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

      2. sqr-pow 97.7%

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

      3. metadata-eval 97.7%

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

      4. metadata-eval 97.7%

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

      5. rem-sqrt-square 97.7%

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

      6. rem-square-sqrt 97.0%

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

      7. fabs-sqr 97.0%

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

      8. rem-square-sqrt 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.9× 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 100.0%

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

   2. Taylor expanded in x around 0 98.0%

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

      1. add-sqr-sqrt_binary64 98.0%

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

   4. Applied rewrite-once 98.0%

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

      1. rem-square-sqrt 98.0%

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

      2. associate--l+ 98.0%

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

   6. Simplified 98.0%

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

   if 1 < x

   1. Initial program 7.5%

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

      1. flip-- 8.3%

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

      2. flip3-+ 8.3%

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

      3. associate-/r/ 8.3%

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

      4. *-commutative 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. +-commutative 8.3%

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

      2. associate-+l- 43.3%

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

      3. +-inverses 43.3%

         \[\leadsto \left(x + \left(1 + \left(x - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)\right)\right) \cdot \frac{1 - \color{blue}{0}}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \]

      4. metadata-eval 43.3%

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

      5. associate-*r/ 43.4%

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

      6. associate-/l* 43.4%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in x around inf 64.2%

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

   7. Taylor expanded in x around inf 97.5%

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

      1. unpow-1 97.5%

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

      2. sqr-pow 97.7%

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

      3. metadata-eval 97.7%

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

      4. metadata-eval 97.7%

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

      5. rem-sqrt-square 97.7%

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

      6. rem-square-sqrt 97.0%

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

      7. fabs-sqr 97.0%

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

      8. rem-square-sqrt 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 97.9%

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

Alternative 5: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 / (1.0 + 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 / (1.0d0 + 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 / (1.0 + 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 / (1.0 + 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(1.0 + 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 / (1.0 + 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[(1.0 + 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 100.0%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. 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. rem-square-sqrt 99.9%

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

      3. div-sub 99.9%

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

      4. sub-neg 99.9%

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

      5. rem-square-sqrt 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. unsub-neg 99.9%

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

      2. div-sub 99.9%

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

      3. +-commutative 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-inverses 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. add-sqr-sqrt_binary64 99.8%

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

   7. Applied rewrite-once 99.8%

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

   8. Taylor expanded in x around 0 96.3%

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

   if 1 < x

   1. Initial program 7.5%

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

      1. flip-- 8.3%

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

      2. flip3-+ 8.3%

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

      3. associate-/r/ 8.3%

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

      4. *-commutative 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. +-commutative 8.3%

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

      2. associate-+l- 43.3%

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

      3. +-inverses 43.3%

         \[\leadsto \left(x + \left(1 + \left(x - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)\right)\right) \cdot \frac{1 - \color{blue}{0}}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \]

      4. metadata-eval 43.3%

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

      5. associate-*r/ 43.4%

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

      6. associate-/l* 43.4%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in x around inf 64.2%

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

   7. Taylor expanded in x around inf 97.5%

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

      1. unpow-1 97.5%

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

      2. sqr-pow 97.7%

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

      3. metadata-eval 97.7%

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

      4. metadata-eval 97.7%

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

      5. rem-sqrt-square 97.7%

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

      6. rem-square-sqrt 97.0%

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

      7. fabs-sqr 97.0%

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

      8. rem-square-sqrt 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 97.0%

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

Alternative 6: 96.9% 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 100.0%

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

   2. Taylor expanded in x around 0 93.6%

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

   if 0.25 < x

   1. Initial program 7.5%

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

      1. flip-- 8.3%

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

      2. flip3-+ 8.3%

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

      3. associate-/r/ 8.3%

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

      4. *-commutative 8.3%

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

   3. Applied egg-rr 8.3%

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

      1. +-commutative 8.3%

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

      2. associate-+l- 43.3%

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

      3. +-inverses 43.3%

         \[\leadsto \left(x + \left(1 + \left(x - \sqrt{\mathsf{fma}\left(x, x, x\right)}\right)\right)\right) \cdot \frac{1 - \color{blue}{0}}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \]

      4. metadata-eval 43.3%

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

      5. associate-*r/ 43.4%

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

      6. associate-/l* 43.4%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in x around inf 64.2%

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

   7. Taylor expanded in x around inf 97.5%

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

      1. unpow-1 97.5%

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

      2. sqr-pow 97.7%

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

      3. metadata-eval 97.7%

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

      4. metadata-eval 97.7%

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

      5. rem-sqrt-square 97.7%

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

      6. rem-square-sqrt 97.0%

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

      7. fabs-sqr 97.0%

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

      8. rem-square-sqrt 97.7%

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

   9. Simplified 97.7%

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

4. Final simplification 95.6%

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

Alternative 7: 9.4% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 7.450580596923828 \cdot 10^{-9} \end{array} \]

FPCore

(FPCore (x) :precision binary64 7.450580596923828e-9)

C

double code(double x) {
	return 7.450580596923828e-9;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 7.450580596923828d-9
end function

Java

public static double code(double x) {
	return 7.450580596923828e-9;
}

Python

def code(x):
	return 7.450580596923828e-9

Julia

function code(x)
	return 7.450580596923828e-9
end

MATLAB

function tmp = code(x)
	tmp = 7.450580596923828e-9;
end

Wolfram

code[x_] := 7.450580596923828e-9

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in x around 0 51.6%

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

4. Applied egg-rr 9.4%

   \[\leadsto \color{blue}{7.450580596923828 \cdot 10^{-9}} \]

5. Final simplification 9.4%

   \[\leadsto 7.450580596923828 \cdot 10^{-9} \]

Alternative 8: 9.8% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 3.814697265625 \cdot 10^{-6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 3.814697265625e-6)

C

double code(double x) {
	return 3.814697265625e-6;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 3.814697265625d-6
end function

Java

public static double code(double x) {
	return 3.814697265625e-6;
}

Python

def code(x):
	return 3.814697265625e-6

Julia

function code(x)
	return 3.814697265625e-6
end

MATLAB

function tmp = code(x)
	tmp = 3.814697265625e-6;
end

Wolfram

code[x_] := 3.814697265625e-6

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in x around 0 51.6%

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

4. Applied egg-rr 9.8%

   \[\leadsto \color{blue}{3.814697265625 \cdot 10^{-6}} \]

5. Final simplification 9.8%

   \[\leadsto 3.814697265625 \cdot 10^{-6} \]

Alternative 9: 10.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.000244140625 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.000244140625)

C

double code(double x) {
	return 0.000244140625;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.000244140625d0
end function

Java

public static double code(double x) {
	return 0.000244140625;
}

Python

def code(x):
	return 0.000244140625

Julia

function code(x)
	return 0.000244140625
end

MATLAB

function tmp = code(x)
	tmp = 0.000244140625;
end

Wolfram

code[x_] := 0.000244140625

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in x around 0 51.6%

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

4. Applied egg-rr 10.2%

   \[\leadsto \color{blue}{0.000244140625} \]

5. Final simplification 10.2%

   \[\leadsto 0.000244140625 \]

Alternative 10: 10.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.015625 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.015625)

C

double code(double x) {
	return 0.015625;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.015625d0
end function

Java

public static double code(double x) {
	return 0.015625;
}

Python

def code(x):
	return 0.015625

Julia

function code(x)
	return 0.015625
end

MATLAB

function tmp = code(x)
	tmp = 0.015625;
end

Wolfram

code[x_] := 0.015625

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in x around 0 51.6%

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

4. Applied egg-rr 10.9%

   \[\leadsto \color{blue}{0.015625} \]

5. Final simplification 10.9%

   \[\leadsto 0.015625 \]

Alternative 11: 13.7% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.75 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.75)

C

double code(double x) {
	return 0.75;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.75d0
end function

Java

public static double code(double x) {
	return 0.75;
}

Python

def code(x):
	return 0.75

Julia

function code(x)
	return 0.75
end

MATLAB

function tmp = code(x)
	tmp = 0.75;
end

Wolfram

code[x_] := 0.75

Derivation

1. Initial program 54.1%

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

2. Taylor expanded in x around 0 51.6%

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

4. Applied egg-rr 13.7%

   \[\leadsto \color{blue}{0.75} \]

5. Final simplification 13.7%

   \[\leadsto 0.75 \]

Alternative 12: 52.1% 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 54.1%

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

2. Taylor expanded in x around 0 50.6%

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

3. Final simplification 50.6%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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