Main:bigenough3 from C

Percentage Accurate: 51.7% → 99.7%

Time: 9.1s

Alternatives: 6

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: 51.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

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 56.8%

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

   1. flip-- 57.2%

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

   2. div-inv 57.2%

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

   3. add-sqr-sqrt 57.5%

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

   4. add-sqr-sqrt 58.2%

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

   5. associate--l+ 58.3%

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

4. Applied egg-rr 58.3%

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

   1. associate-*r/ 58.3%

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

   2. *-rgt-identity 58.3%

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

   3. +-commutative 58.3%

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

   4. associate-+l- 99.7%

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

   5. +-inverses 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 4e-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 <= 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((1.0d0 + x)) - 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((1.0 + x)) - 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((1.0 + x)) - 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(1.0 + x)) - 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((1.0 + x)) - 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[(1.0 + x), $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 5.4%

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

      1. expm1-log1p-u 5.4%

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

   4. Applied egg-rr 5.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 5.4%

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

      2. +-commutative 5.4%

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

      3. flip-- 6.0%

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

      4. flip3-+ 6.0%

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

      5. associate-/r/ 6.0%

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

   6. Applied egg-rr 5.8%

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

      1. +-commutative 5.8%

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

      2. associate-+l- 46.4%

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

      3. +-inverses 46.4%

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

      4. metadata-eval 46.4%

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

      5. associate-*l/ 46.4%

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

      6. *-lft-identity 46.4%

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

      7. associate-+l+ 46.4%

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

      8. *-commutative 46.4%

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

      9. +-commutative 46.4%

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

      10. +-commutative 46.4%

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

   8. Simplified 46.4%

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

   9. Taylor expanded in x around inf 61.8%

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

       1. associate-+r+ 61.8%

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

       2. +-commutative 61.8%

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

       3. associate-*r/ 61.8%

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

       4. metadata-eval 61.8%

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

   11. Simplified 61.8%

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

   12. Taylor expanded in x around inf 99.2%

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

       1. unpow-1 99.2%

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

       2. sqr-pow 99.3%

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

       3. metadata-eval 99.3%

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

       4. metadata-eval 99.3%

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

       5. rem-sqrt-square 99.3%

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

       6. rem-square-sqrt 98.5%

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

       7. fabs-sqr 98.5%

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

       8. rem-square-sqrt 99.3%

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

   14. Simplified 99.3%

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

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

   1. Initial program 99.4%

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

4. Final simplification 99.4%

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

Alternative 3: 96.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 0.38) {
		tmp = 1.0 / (1.0 + (1.0 + (pow(x, 1.5) + -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 <= 0.38d0) then
        tmp = 1.0d0 / (1.0d0 + (1.0d0 + ((x ** 1.5d0) + (-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 <= 0.38) {
		tmp = 1.0 / (1.0 + (1.0 + (Math.pow(x, 1.5) + -1.0)));
	} 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 + (1.0 + (math.pow(x, 1.5) + -1.0)))
	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 + Float64(1.0 + Float64((x ^ 1.5) + -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 <= 0.38)
		tmp = 1.0 / (1.0 + (1.0 + ((x ^ 1.5) + -1.0)));
	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[(1.0 + N[(N[Power[x, 1.5], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.38

   1. Initial program 99.9%

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

      1. expm1-log1p-u 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 99.9%

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

      2. +-commutative 99.9%

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

      3. flip-- 99.9%

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

      4. flip3-+ 99.8%

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

      5. associate-/r/ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l- 99.9%

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

      3. +-inverses 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-*l/ 99.9%

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

      6. *-lft-identity 99.9%

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

      7. associate-+l+ 99.9%

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

      8. *-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 94.6%

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

       1. expm1-log1p-u 94.6%

          \[\leadsto \frac{1}{1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{1.5}\right)\right)}} \]

       2. log1p-define 94.6%

          \[\leadsto \frac{1}{1 + \mathsf{expm1}\left(\color{blue}{\log \left(1 + {x}^{1.5}\right)}\right)} \]

       3. expm1-undefine 94.6%

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

       4. add-exp-log 94.6%

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

   11. Applied egg-rr 94.6%

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

       1. associate--l+ 94.6%

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

   13. Simplified 94.6%

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

   if 0.38 < x

   1. Initial program 8.7%

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

      1. expm1-log1p-u 8.7%

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

   4. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 8.7%

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

      2. +-commutative 8.7%

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

      3. flip-- 9.7%

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

      4. flip3-+ 9.7%

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

      5. associate-/r/ 9.7%

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

   6. Applied egg-rr 9.7%

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

      1. +-commutative 9.7%

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

      2. associate-+l- 48.6%

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

      3. +-inverses 48.6%

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

      4. metadata-eval 48.6%

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

      5. associate-*l/ 48.6%

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

      6. *-lft-identity 48.6%

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

      7. associate-+l+ 48.6%

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

      8. *-commutative 48.6%

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

      9. +-commutative 48.6%

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

      10. +-commutative 48.6%

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

   8. Simplified 48.6%

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

   9. Taylor expanded in x around inf 62.0%

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

       1. associate-+r+ 62.0%

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

       2. +-commutative 62.0%

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

       3. associate-*r/ 62.0%

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

       4. metadata-eval 62.0%

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

   11. Simplified 62.0%

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

   12. Taylor expanded in x around inf 96.7%

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

       1. unpow-1 96.7%

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

       2. sqr-pow 96.8%

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

       3. metadata-eval 96.8%

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

       4. metadata-eval 96.8%

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

       5. rem-sqrt-square 96.8%

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

       6. rem-square-sqrt 96.1%

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

       7. fabs-sqr 96.1%

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

       8. rem-square-sqrt 96.8%

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

   14. Simplified 96.8%

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

4. Final simplification 95.7%

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

Alternative 4: 96.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.38

   1. Initial program 99.9%

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

      1. expm1-log1p-u 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 99.9%

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

      2. +-commutative 99.9%

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

      3. flip-- 99.9%

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

      4. flip3-+ 99.8%

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

      5. associate-/r/ 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l- 99.9%

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

      3. +-inverses 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-*l/ 99.9%

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

      6. *-lft-identity 99.9%

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

      7. associate-+l+ 99.9%

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

      8. *-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in x around 0 94.6%

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

   if 0.38 < x

   1. Initial program 8.7%

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

      1. expm1-log1p-u 8.7%

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

   4. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 8.7%

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

      2. +-commutative 8.7%

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

      3. flip-- 9.7%

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

      4. flip3-+ 9.7%

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

      5. associate-/r/ 9.7%

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

   6. Applied egg-rr 9.7%

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

      1. +-commutative 9.7%

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

      2. associate-+l- 48.6%

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

      3. +-inverses 48.6%

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

      4. metadata-eval 48.6%

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

      5. associate-*l/ 48.6%

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

      6. *-lft-identity 48.6%

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

      7. associate-+l+ 48.6%

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

      8. *-commutative 48.6%

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

      9. +-commutative 48.6%

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

      10. +-commutative 48.6%

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

   8. Simplified 48.6%

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

   9. Taylor expanded in x around inf 62.0%

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

       1. associate-+r+ 62.0%

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

       2. +-commutative 62.0%

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

       3. associate-*r/ 62.0%

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

       4. metadata-eval 62.0%

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

   11. Simplified 62.0%

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

   12. Taylor expanded in x around inf 96.7%

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

       1. unpow-1 96.7%

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

       2. sqr-pow 96.8%

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

       3. metadata-eval 96.8%

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

       4. metadata-eval 96.8%

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

       5. rem-sqrt-square 96.8%

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

       6. rem-square-sqrt 96.1%

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

       7. fabs-sqr 96.1%

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

       8. rem-square-sqrt 96.8%

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

   14. Simplified 96.8%

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

4. Final simplification 95.7%

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

Alternative 5: 96.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 95.2%

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

   if 0.25 < x

   1. Initial program 9.5%

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

      1. expm1-log1p-u 9.5%

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

   4. Applied egg-rr 9.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x + 1} - \sqrt{x}\right)\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 9.5%

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

      2. +-commutative 9.5%

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

      3. flip-- 10.4%

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

      4. flip3-+ 10.4%

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

      5. associate-/r/ 10.4%

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

   6. Applied egg-rr 10.4%

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

      1. +-commutative 10.4%

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

      2. associate-+l- 49.0%

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

      3. +-inverses 49.0%

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

      4. metadata-eval 49.0%

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

      5. associate-*l/ 49.1%

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

      6. *-lft-identity 49.1%

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

      7. associate-+l+ 49.1%

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

      8. *-commutative 49.1%

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

      9. +-commutative 49.1%

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

      10. +-commutative 49.1%

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

   8. Simplified 49.1%

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

   9. Taylor expanded in x around inf 61.7%

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

       1. associate-+r+ 61.7%

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

       2. +-commutative 61.7%

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

       3. associate-*r/ 61.7%

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

       4. metadata-eval 61.7%

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

   11. Simplified 61.7%

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

   12. Taylor expanded in x around inf 96.0%

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

       1. unpow-1 96.0%

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

       2. sqr-pow 96.2%

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

       3. metadata-eval 96.2%

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

       4. metadata-eval 96.2%

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

       5. rem-sqrt-square 96.2%

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

       6. rem-square-sqrt 95.4%

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

       7. fabs-sqr 95.4%

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

       8. rem-square-sqrt 96.2%

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

   14. Simplified 96.2%

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

4. Final simplification 95.7%

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

Alternative 6: 49.6% 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 56.8%

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

3. Taylor expanded in x around 0 53.2%

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

4. Final simplification 53.2%

   \[\leadsto 1 \]
5. Add Preprocessing

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 2024034 
(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)))