Main:bigenough3 from C

Percentage Accurate: 53.4% → 99.7%

Time: 6.5s

Alternatives: 7

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

   1. flip-- 52.9%

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

   2. div-inv 52.9%

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

   3. add-sqr-sqrt 52.6%

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

   4. add-sqr-sqrt 53.7%

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

3. Applied egg-rr 53.7%

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

   1. associate-*r/ 53.7%

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

   2. *-rgt-identity 53.7%

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

   3. remove-double-neg 53.7%

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

   4. sub-neg 53.7%

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

   5. div-sub 52.1%

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

   6. rem-square-sqrt 52.1%

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

   7. sqr-neg 52.1%

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

   8. div-sub 52.6%

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

   9. +-commutative 52.6%

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

   10. sqr-neg 52.6%

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

   11. rem-square-sqrt 53.7%

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

   12. associate--l+ 99.7%

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

   13. +-inverses 99.7%

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

   14. metadata-eval 99.7%

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

   15. sub-neg 99.7%

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

   16. remove-double-neg 99.7%

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

5. Simplified 99.7%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 99.7%

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

   3. inv-pow 99.7%

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

   4. inv-pow 99.7%

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

   5. pow-prod-up 99.7%

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

   6. +-commutative 99.7%

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

   7. metadata-eval 99.7%

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

7. Applied egg-rr 99.7%

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

   1. sqrt-pow1 99.7%

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

   2. metadata-eval 99.7%

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

   3. metadata-eval 99.7%

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

   4. pow-prod-up 99.4%

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

   5. pow-prod-down 99.7%

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

   6. pow2 99.7%

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 4e-5) (* 0.5 (pow x -0.5)) t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 4e-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((x + 1.0d0)) - sqrt(x)
    if (t_0 <= 4d-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((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 4e-5) {
		tmp = 0.5 * Math.pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 4e-5:
		tmp = 0.5 * math.pow(x, -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 4e-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((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 4e-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[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-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)) < 4.00000000000000033e-5

   1. Initial program 5.2%

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

      1. add-log-exp 5.2%

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

   3. Applied egg-rr 5.2%

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

      1. add-log-exp 5.2%

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

      2. flip-- 6.8%

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

      3. +-commutative 6.8%

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

      4. flip3-+ 6.8%

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

      5. associate-/r/ 6.8%

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

   5. Applied egg-rr 6.3%

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

      1. +-commutative 6.3%

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

      2. associate-+l- 53.4%

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

      3. +-inverses 53.4%

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

      4. metadata-eval 53.4%

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

      5. associate-*l/ 53.5%

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

   7. Simplified 53.5%

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

   8. Taylor expanded in x around inf 77.4%

      \[\leadsto \frac{\color{blue}{0.5 + \left(x + 0.125 \cdot \frac{1}{x}\right)}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]
   9. Step-by-step derivation

      1. associate-+r+ 77.4%

         \[\leadsto \frac{\color{blue}{\left(0.5 + x\right) + 0.125 \cdot \frac{1}{x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

      2. associate-*r/ 77.4%

         \[\leadsto \frac{\left(0.5 + x\right) + \color{blue}{\frac{0.125 \cdot 1}{x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

      3. metadata-eval 77.4%

         \[\leadsto \frac{\left(0.5 + x\right) + \frac{\color{blue}{0.125}}{x}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

   10. Simplified 77.4%

       \[\leadsto \frac{\color{blue}{\left(0.5 + x\right) + \frac{0.125}{x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

   11. Taylor expanded in x around inf 99.2%

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

       1. unpow-1 99.2%

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

       2. metadata-eval 99.2%

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

       3. pow-sqr 99.4%

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

       4. rem-sqrt-square 99.4%

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

       5. rem-square-sqrt 98.7%

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

       6. fabs-sqr 98.7%

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

       7. rem-square-sqrt 99.4%

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

   13. Simplified 99.4%

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

   if 4.00000000000000033e-5 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.7%

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

4. Final simplification 99.6%

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

Alternative 3: 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]

Derivation

1. Initial program 52.1%

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

   1. flip-- 52.9%

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

   2. div-inv 52.9%

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

   3. add-sqr-sqrt 52.6%

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

   4. add-sqr-sqrt 53.7%

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

3. Applied egg-rr 53.7%

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

   1. associate-*r/ 53.7%

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

   2. *-rgt-identity 53.7%

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

   3. remove-double-neg 53.7%

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

   4. sub-neg 53.7%

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

   5. div-sub 52.1%

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

   6. rem-square-sqrt 52.1%

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

   7. sqr-neg 52.1%

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

   8. div-sub 52.6%

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

   9. +-commutative 52.6%

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

   10. sqr-neg 52.6%

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

   11. rem-square-sqrt 53.7%

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

   12. associate--l+ 99.7%

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

   13. +-inverses 99.7%

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

   14. metadata-eval 99.7%

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

   15. sub-neg 99.7%

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

   16. remove-double-neg 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 98.7% 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 99.9%

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

   2. Taylor expanded in x around 0 99.9%

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

      1. associate--l+ 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

   if 1 < x

   1. Initial program 5.7%

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

      1. add-log-exp 5.7%

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

   3. Applied egg-rr 5.7%

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

      1. add-log-exp 5.7%

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

      2. flip-- 7.3%

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

      3. +-commutative 7.3%

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

      4. flip3-+ 7.3%

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

      5. associate-/r/ 7.3%

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

   5. Applied egg-rr 7.1%

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

      1. +-commutative 7.1%

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

      2. associate-+l- 53.8%

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

      3. +-inverses 53.8%

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

      4. metadata-eval 53.8%

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

      5. associate-*l/ 53.8%

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

   7. Simplified 53.8%

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

   8. Taylor expanded in x around inf 77.5%

      \[\leadsto \frac{\color{blue}{0.5 + \left(x + 0.125 \cdot \frac{1}{x}\right)}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]
   9. Step-by-step derivation

      1. associate-+r+ 77.5%

         \[\leadsto \frac{\color{blue}{\left(0.5 + x\right) + 0.125 \cdot \frac{1}{x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

      2. associate-*r/ 77.5%

         \[\leadsto \frac{\left(0.5 + x\right) + \color{blue}{\frac{0.125 \cdot 1}{x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

      3. metadata-eval 77.5%

         \[\leadsto \frac{\left(0.5 + x\right) + \frac{\color{blue}{0.125}}{x}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

   10. Simplified 77.5%

       \[\leadsto \frac{\color{blue}{\left(0.5 + x\right) + \frac{0.125}{x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

   11. Taylor expanded in x around inf 98.8%

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

       1. unpow-1 98.8%

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

       2. metadata-eval 98.8%

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

       3. pow-sqr 99.0%

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

       4. rem-sqrt-square 99.0%

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

       5. rem-square-sqrt 98.3%

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

       6. fabs-sqr 98.3%

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

       7. rem-square-sqrt 99.0%

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

   13. Simplified 99.0%

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

4. Final simplification 99.5%

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

Alternative 5: 97.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.38

   1. Initial program 99.9%

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

      1. add-log-exp 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. add-log-exp 99.9%

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

      2. flip-- 99.8%

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

      3. +-commutative 99.8%

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

      4. flip3-+ 99.8%

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

      5. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l- 99.9%

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

      3. +-inverses 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-*l/ 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 95.4%

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

   if 0.38 < x

   1. Initial program 5.7%

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

      1. add-log-exp 5.7%

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

   3. Applied egg-rr 5.7%

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

      1. add-log-exp 5.7%

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

      2. flip-- 7.3%

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

      3. +-commutative 7.3%

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

      4. flip3-+ 7.3%

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

      5. associate-/r/ 7.3%

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

   5. Applied egg-rr 7.1%

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

      1. +-commutative 7.1%

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

      2. associate-+l- 53.8%

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

      3. +-inverses 53.8%

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

      4. metadata-eval 53.8%

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

      5. associate-*l/ 53.8%

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

   7. Simplified 53.8%

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

   8. Taylor expanded in x around inf 77.5%

      \[\leadsto \frac{\color{blue}{0.5 + \left(x + 0.125 \cdot \frac{1}{x}\right)}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]
   9. Step-by-step derivation

      1. associate-+r+ 77.5%

         \[\leadsto \frac{\color{blue}{\left(0.5 + x\right) + 0.125 \cdot \frac{1}{x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

      2. associate-*r/ 77.5%

         \[\leadsto \frac{\left(0.5 + x\right) + \color{blue}{\frac{0.125 \cdot 1}{x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

      3. metadata-eval 77.5%

         \[\leadsto \frac{\left(0.5 + x\right) + \frac{\color{blue}{0.125}}{x}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

   10. Simplified 77.5%

       \[\leadsto \frac{\color{blue}{\left(0.5 + x\right) + \frac{0.125}{x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

   11. Taylor expanded in x around inf 98.8%

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

       1. unpow-1 98.8%

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

       2. metadata-eval 98.8%

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

       3. pow-sqr 99.0%

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

       4. rem-sqrt-square 99.0%

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

       5. rem-square-sqrt 98.3%

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

       6. fabs-sqr 98.3%

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

       7. rem-square-sqrt 99.0%

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

   13. Simplified 99.0%

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

4. Final simplification 97.2%

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

Alternative 6: 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 99.9%

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

   2. Taylor expanded in x around 0 95.4%

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

   if 0.25 < x

   1. Initial program 5.7%

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

      1. add-log-exp 5.7%

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

   3. Applied egg-rr 5.7%

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

      1. add-log-exp 5.7%

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

      2. flip-- 7.3%

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

      3. +-commutative 7.3%

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

      4. flip3-+ 7.3%

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

      5. associate-/r/ 7.3%

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

   5. Applied egg-rr 7.1%

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

      1. +-commutative 7.1%

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

      2. associate-+l- 53.8%

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

      3. +-inverses 53.8%

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

      4. metadata-eval 53.8%

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

      5. associate-*l/ 53.8%

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

   7. Simplified 53.8%

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

   8. Taylor expanded in x around inf 77.5%

      \[\leadsto \frac{\color{blue}{0.5 + \left(x + 0.125 \cdot \frac{1}{x}\right)}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]
   9. Step-by-step derivation

      1. associate-+r+ 77.5%

         \[\leadsto \frac{\color{blue}{\left(0.5 + x\right) + 0.125 \cdot \frac{1}{x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

      2. associate-*r/ 77.5%

         \[\leadsto \frac{\left(0.5 + x\right) + \color{blue}{\frac{0.125 \cdot 1}{x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

      3. metadata-eval 77.5%

         \[\leadsto \frac{\left(0.5 + x\right) + \frac{\color{blue}{0.125}}{x}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

   10. Simplified 77.5%

       \[\leadsto \frac{\color{blue}{\left(0.5 + x\right) + \frac{0.125}{x}}}{{\left(1 + x\right)}^{1.5} + {x}^{1.5}} \]

   11. Taylor expanded in x around inf 98.8%

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

       1. unpow-1 98.8%

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

       2. metadata-eval 98.8%

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

       3. pow-sqr 99.0%

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

       4. rem-sqrt-square 99.0%

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

       5. rem-square-sqrt 98.3%

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

       6. fabs-sqr 98.3%

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

       7. rem-square-sqrt 99.0%

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

   13. Simplified 99.0%

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

4. Final simplification 97.2%

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

Alternative 7: 51.5% 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 52.1%

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

2. Taylor expanded in x around 0 50.5%

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

3. Final simplification 50.5%

   \[\leadsto 1 \]

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 2023272 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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