2sqrt (example 3.1)

Percentage Accurate: 53.5% → 99.7%

Time: 8.2s

Alternatives: 8

Speedup: 2.0×

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: 53.5% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left({\left(\sqrt{1 + x} + \sqrt{x}\right)}^{2}\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (pow (+ (sqrt (+ 1.0 x)) (sqrt x)) 2.0) -0.5))

C

double code(double x) {
	return pow(pow((sqrt((1.0 + x)) + sqrt(x)), 2.0), -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((sqrt((1.0d0 + x)) + sqrt(x)) ** 2.0d0) ** (-0.5d0)
end function

Java

public static double code(double x) {
	return Math.pow(Math.pow((Math.sqrt((1.0 + x)) + Math.sqrt(x)), 2.0), -0.5);
}

Python

def code(x):
	return math.pow(math.pow((math.sqrt((1.0 + x)) + math.sqrt(x)), 2.0), -0.5)

Julia

function code(x)
	return (Float64(sqrt(Float64(1.0 + x)) + sqrt(x)) ^ 2.0) ^ -0.5
end

MATLAB

function tmp = code(x)
	tmp = ((sqrt((1.0 + x)) + sqrt(x)) ^ 2.0) ^ -0.5;
end

Wolfram

code[x_] := N[Power[N[Power[N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], -0.5], $MachinePrecision]

Derivation

1. Initial program 50.0%

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

   1. flip-- 50.0%

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

   2. div-inv 50.0%

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

   3. add-sqr-sqrt 50.5%

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

   4. add-sqr-sqrt 50.7%

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

   5. associate--l+ 50.7%

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

3. Applied egg-rr 50.7%

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

   1. associate-*r/ 50.7%

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

   2. *-rgt-identity 50.7%

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

   3. +-commutative 50.7%

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

   4. associate-+l- 99.8%

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

   5. +-inverses 99.8%

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

   6. metadata-eval 99.8%

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

   7. +-commutative 99.8%

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

5. Simplified 99.8%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 99.8%

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

   3. inv-pow 99.8%

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

   4. inv-pow 99.8%

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

   5. pow-prod-up 99.7%

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

   6. metadata-eval 99.7%

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

7. Applied egg-rr 99.7%

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

   1. sqrt-pow1 99.8%

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

   2. metadata-eval 99.8%

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

   3. metadata-eval 99.8%

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

   4. pow-prod-up 99.5%

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

   5. pow-prod-down 99.8%

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

   6. pow2 99.8%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 2e-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 <= 2e-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 <= 2d-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 <= 2e-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 <= 2e-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 <= 2e-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 <= 2e-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, 2e-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)) < 2.00000000000000016e-5

   1. Initial program 4.5%

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

      1. flip-- 4.5%

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

      2. div-inv 4.5%

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

      3. add-sqr-sqrt 5.1%

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

      4. add-sqr-sqrt 5.3%

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

      5. associate--l+ 5.3%

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

   3. Applied egg-rr 5.3%

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

      1. associate-*r/ 5.3%

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

      2. *-rgt-identity 5.3%

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

      3. +-commutative 5.3%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. flip3-+ 65.8%

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

      2. associate-/r/ 65.8%

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

      3. pow1/2 65.8%

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

      4. +-commutative 65.8%

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

      5. pow-pow 65.8%

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

      6. metadata-eval 65.8%

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

      7. +-commutative 65.8%

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

      8. sqrt-pow2 65.7%

         \[\leadsto \frac{1}{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      9. metadata-eval 65.7%

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

      10. add-sqr-sqrt 65.9%

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

      11. +-commutative 65.9%

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

      12. add-sqr-sqrt 65.6%

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

   7. Applied egg-rr 49.5%

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

   8. Taylor expanded in x around inf 99.4%

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

      1. expm1-log1p-u 99.4%

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

      2. expm1-udef 5.6%

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

      3. inv-pow 5.6%

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

      4. sqrt-pow1 5.6%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 5.6%

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

   10. Applied egg-rr 5.6%

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

       1. expm1-def 99.6%

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

       2. expm1-log1p 99.6%

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

   12. Simplified 99.6%

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

   if 2.00000000000000016e-5 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.0%

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

   1. flip-- 50.0%

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

   2. div-inv 50.0%

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

   3. add-sqr-sqrt 50.5%

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

   4. add-sqr-sqrt 50.7%

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

   5. associate--l+ 50.7%

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

3. Applied egg-rr 50.7%

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

   1. associate-*r/ 50.7%

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

   2. *-rgt-identity 50.7%

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

   3. +-commutative 50.7%

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

   4. associate-+l- 99.8%

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

   5. +-inverses 99.8%

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

   6. metadata-eval 99.8%

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

   7. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.74:\\ \;\;\;\;\frac{1}{\frac{1}{1 - {x}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.74) (/ 1.0 (/ 1.0 (- 1.0 (pow x 1.5)))) (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.74) {
		tmp = 1.0 / (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 <= 0.74d0) then
        tmp = 1.0d0 / (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 <= 0.74) {
		tmp = 1.0 / (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 <= 0.74:
		tmp = 1.0 / (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 <= 0.74)
		tmp = Float64(1.0 / 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 <= 0.74)
		tmp = 1.0 / (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, 0.74], N[(1.0 / N[(1.0 / N[(1.0 - N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.73999999999999999

   1. Initial program 100.0%

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

      1. flip3-- 99.9%

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

      2. clear-num 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate-+l+ 99.9%

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

      5. add-sqr-sqrt 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-def 99.9%

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

      8. sqrt-pow2 99.9%

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

      9. metadata-eval 99.9%

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

      10. sqrt-pow2 99.9%

          \[\leadsto \frac{1}{\frac{x + \left(1 + \mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)\right)}{{\left(x + 1\right)}^{1.5} - \color{blue}{{x}^{\left(\frac{3}{2}\right)}}}} \]

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 96.8%

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

   if 0.73999999999999999 < x

   1. Initial program 7.8%

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

      1. flip-- 8.0%

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

      2. div-inv 8.0%

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

      3. add-sqr-sqrt 8.9%

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

      4. add-sqr-sqrt 9.4%

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

      5. associate--l+ 9.4%

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

   3. Applied egg-rr 9.4%

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

      1. associate-*r/ 9.4%

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

      2. *-rgt-identity 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. flip3-+ 67.2%

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

      2. associate-/r/ 67.2%

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

      3. pow1/2 67.2%

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

      4. +-commutative 67.2%

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

      5. pow-pow 67.2%

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

      6. metadata-eval 67.2%

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

      7. +-commutative 67.2%

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

      8. sqrt-pow2 67.1%

         \[\leadsto \frac{1}{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      9. metadata-eval 67.1%

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

      10. add-sqr-sqrt 67.3%

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

      11. +-commutative 67.3%

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

      12. add-sqr-sqrt 67.1%

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

   7. Applied egg-rr 51.7%

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

   8. Taylor expanded in x around inf 96.9%

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

      1. expm1-log1p-u 96.9%

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

      2. expm1-udef 7.2%

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

      3. inv-pow 7.2%

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

      4. sqrt-pow1 7.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 7.2%

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

   10. Applied egg-rr 7.2%

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

       1. expm1-def 97.1%

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

       2. expm1-log1p 97.1%

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

   12. Simplified 97.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.74:\\ \;\;\;\;\frac{1}{\frac{1}{1 - {x}^{1.5}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 5: 97.0% 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 96.8%

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

   if 0.25 < x

   1. Initial program 7.8%

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

      1. flip-- 8.0%

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

      2. div-inv 8.0%

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

      3. add-sqr-sqrt 8.9%

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

      4. add-sqr-sqrt 9.4%

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

      5. associate--l+ 9.4%

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

   3. Applied egg-rr 9.4%

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

      1. associate-*r/ 9.4%

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

      2. *-rgt-identity 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. flip3-+ 67.2%

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

      2. associate-/r/ 67.2%

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

      3. pow1/2 67.2%

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

      4. +-commutative 67.2%

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

      5. pow-pow 67.2%

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

      6. metadata-eval 67.2%

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

      7. +-commutative 67.2%

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

      8. sqrt-pow2 67.1%

         \[\leadsto \frac{1}{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      9. metadata-eval 67.1%

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

      10. add-sqr-sqrt 67.3%

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

      11. +-commutative 67.3%

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

      12. add-sqr-sqrt 67.1%

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

   7. Applied egg-rr 51.7%

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

   8. Taylor expanded in x around inf 96.9%

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

      1. expm1-log1p-u 96.9%

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

      2. expm1-udef 7.2%

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

      3. inv-pow 7.2%

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

      4. sqrt-pow1 7.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 7.2%

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

   10. Applied egg-rr 7.2%

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

       1. expm1-def 97.1%

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

       2. expm1-log1p 97.1%

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

   12. Simplified 97.1%

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

4. Final simplification 97.0%

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

Alternative 6: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.74:\\ \;\;\;\;1 - {x}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.74) (- 1.0 (pow x 1.5)) (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.74) {
		tmp = 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 <= 0.74d0) then
        tmp = 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 <= 0.74) {
		tmp = 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 <= 0.74:
		tmp = 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 <= 0.74)
		tmp = 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 <= 0.74)
		tmp = 1.0 - (x ^ 1.5);
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.74], N[(1.0 - N[Power[x, 1.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.73999999999999999

   1. Initial program 100.0%

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

      1. flip3-- 99.9%

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

      2. clear-num 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate-+l+ 99.9%

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

      5. add-sqr-sqrt 99.9%

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

      6. +-commutative 99.9%

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

      7. fma-def 99.9%

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

      8. sqrt-pow2 99.9%

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

      9. metadata-eval 99.9%

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

      10. sqrt-pow2 99.9%

          \[\leadsto \frac{1}{\frac{x + \left(1 + \mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)\right)}{{\left(x + 1\right)}^{1.5} - \color{blue}{{x}^{\left(\frac{3}{2}\right)}}}} \]

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 96.8%

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

   5. Taylor expanded in x around 0 96.8%

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

   if 0.73999999999999999 < x

   1. Initial program 7.8%

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

      1. flip-- 8.0%

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

      2. div-inv 8.0%

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

      3. add-sqr-sqrt 8.9%

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

      4. add-sqr-sqrt 9.4%

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

      5. associate--l+ 9.4%

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

   3. Applied egg-rr 9.4%

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

      1. associate-*r/ 9.4%

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

      2. *-rgt-identity 9.4%

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

      3. +-commutative 9.4%

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

      4. associate-+l- 99.6%

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

      5. +-inverses 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   5. Simplified 99.6%

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

      1. flip3-+ 67.2%

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

      2. associate-/r/ 67.2%

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

      3. pow1/2 67.2%

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

      4. +-commutative 67.2%

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

      5. pow-pow 67.2%

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

      6. metadata-eval 67.2%

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

      7. +-commutative 67.2%

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

      8. sqrt-pow2 67.1%

         \[\leadsto \frac{1}{{\left(1 + x\right)}^{1.5} + \color{blue}{{x}^{\left(\frac{3}{2}\right)}}} \cdot \left(\sqrt{1 + x} \cdot \sqrt{1 + x} + \left(\sqrt{x} \cdot \sqrt{x} - \sqrt{1 + x} \cdot \sqrt{x}\right)\right) \]

      9. metadata-eval 67.1%

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

      10. add-sqr-sqrt 67.3%

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

      11. +-commutative 67.3%

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

      12. add-sqr-sqrt 67.1%

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

   7. Applied egg-rr 51.7%

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

   8. Taylor expanded in x around inf 96.9%

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

      1. expm1-log1p-u 96.9%

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

      2. expm1-udef 7.2%

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

      3. inv-pow 7.2%

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

      4. sqrt-pow1 7.2%

         \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 7.2%

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

   10. Applied egg-rr 7.2%

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

       1. expm1-def 97.1%

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

       2. expm1-log1p 97.1%

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

   12. Simplified 97.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.74:\\ \;\;\;\;1 - {x}^{1.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 7: 58.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5625}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.56) 1.0 (sqrt (/ 0.5625 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.56) {
		tmp = 1.0;
	} else {
		tmp = sqrt((0.5625 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.56d0) then
        tmp = 1.0d0
    else
        tmp = sqrt((0.5625d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.56) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt((0.5625 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.56:
		tmp = 1.0
	else:
		tmp = math.sqrt((0.5625 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.56)
		tmp = 1.0;
	else
		tmp = sqrt(Float64(0.5625 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.56)
		tmp = 1.0;
	else
		tmp = sqrt((0.5625 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.56], 1.0, N[Sqrt[N[(0.5625 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.56000000000000005

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 96.8%

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

   if 0.56000000000000005 < x

   1. Initial program 7.8%

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

      1. flip3-- 6.4%

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

      2. clear-num 6.4%

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

      3. add-sqr-sqrt 6.4%

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

      4. associate-+l+ 6.4%

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

      5. add-sqr-sqrt 6.4%

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

      6. +-commutative 6.4%

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

      7. fma-def 6.3%

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

      8. sqrt-pow2 7.0%

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

      9. metadata-eval 7.0%

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

      10. sqrt-pow2 6.5%

          \[\leadsto \frac{1}{\frac{x + \left(1 + \mathsf{fma}\left(\sqrt{x + 1}, \sqrt{x}, x\right)\right)}{{\left(x + 1\right)}^{1.5} - \color{blue}{{x}^{\left(\frac{3}{2}\right)}}}} \]

   3. Applied egg-rr 6.5%

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

   4. Taylor expanded in x around inf 20.0%

      \[\leadsto \color{blue}{0.75 \cdot \sqrt{\frac{1}{x}}} \]
   5. Step-by-step derivation

      1. *-commutative 20.0%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.75} \]

   6. Simplified 20.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot 0.75} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 20.0%

         \[\leadsto \color{blue}{\sqrt{\sqrt{\frac{1}{x}} \cdot 0.75} \cdot \sqrt{\sqrt{\frac{1}{x}} \cdot 0.75}} \]

      2. sqrt-unprod 20.0%

         \[\leadsto \color{blue}{\sqrt{\left(\sqrt{\frac{1}{x}} \cdot 0.75\right) \cdot \left(\sqrt{\frac{1}{x}} \cdot 0.75\right)}} \]

      3. swap-sqr 20.0%

         \[\leadsto \sqrt{\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \sqrt{\frac{1}{x}}\right) \cdot \left(0.75 \cdot 0.75\right)}} \]

      4. add-sqr-sqrt 20.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1}{x}} \cdot \left(0.75 \cdot 0.75\right)} \]

      5. metadata-eval 20.0%

         \[\leadsto \sqrt{\frac{1}{x} \cdot \color{blue}{0.5625}} \]

   8. Applied egg-rr 20.0%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x} \cdot 0.5625}} \]
   9. Step-by-step derivation

      1. associate-*l/ 20.0%

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 0.5625}{x}}} \]

      2. metadata-eval 20.0%

         \[\leadsto \sqrt{\frac{\color{blue}{0.5625}}{x}} \]

   10. Simplified 20.0%

       \[\leadsto \color{blue}{\sqrt{\frac{0.5625}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.56:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5625}{x}}\\ \end{array} \]

Alternative 8: 51.6% 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. Taylor expanded in x around 0 48.1%

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

3. Final simplification 48.1%

   \[\leadsto 1 \]

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 2023313 
(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)))