Main:bigenough3 from C

Percentage Accurate: 52.6% → 99.8%

Time: 9.1s

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: 52.6% 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: 99.8% 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 49.1%

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

   1. flip-- 49.0%

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

   2. div-inv 49.0%

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

   3. add-sqr-sqrt 49.3%

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

   4. add-sqr-sqrt 49.6%

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

   5. associate--l+ 49.6%

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

4. Applied egg-rr 49.6%

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

   1. associate-*r/ 49.6%

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

   2. *-rgt-identity 49.6%

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

   3. +-commutative 49.6%

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

   4. associate-+l- 99.7%

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

   5. +-inverses 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 2e-6) (* 0.5 (sqrt (/ 1.0 x))) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 2e-6) {
		tmp = 0.5 * sqrt((1.0 / x));
	} 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-6) then
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    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-6) {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	} 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-6:
		tmp = 0.5 * math.sqrt((1.0 / x))
	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-6)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	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-6)
		tmp = 0.5 * sqrt((1.0 / x));
	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-6], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 1.99999999999999991e-6

   1. Initial program 4.3%

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

      1. flip-- 4.3%

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

      2. div-inv 4.3%

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

      3. add-sqr-sqrt 4.7%

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

      4. add-sqr-sqrt 5.2%

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

      5. associate--l+ 5.2%

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

   4. Applied egg-rr 5.2%

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

      1. associate-*r/ 5.2%

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

      2. *-rgt-identity 5.2%

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

      3. +-commutative 5.2%

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

      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%

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

   8. Applied egg-rr 50.6%

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

      1. associate-*l/ 50.7%

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

      2. *-lft-identity 50.7%

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

      3. associate-+r+ 50.7%

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

      4. associate--l+ 50.7%

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

      5. count-2 50.7%

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

      6. distribute-lft1-in 50.7%

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

      7. unpow1 50.7%

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

      8. sqr-pow 50.7%

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

      9. hypot-def 68.1%

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

      10. metadata-eval 68.1%

          \[\leadsto \frac{2 \cdot x + \left(1 - \mathsf{hypot}\left(x, {x}^{\color{blue}{0.5}}\right)\right)}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \]

      11. unpow1/2 68.1%

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

      12. +-commutative 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in x around inf 99.6%

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

   if 1.99999999999999991e-6 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.31:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, 1\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.31) (pow (fma x 2.0 1.0) -2.0) (* 0.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.31) {
		tmp = pow(fma(x, 2.0, 1.0), -2.0);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.31)
		tmp = fma(x, 2.0, 1.0) ^ -2.0;
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.31], N[Power[N[(x * 2.0 + 1.0), $MachinePrecision], -2.0], $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.309999999999999998

   1. Initial program 99.9%

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

      1. flip-- 99.8%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l- 99.9%

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

      5. +-inverses 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 99.9%

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

      3. frac-times 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. sqr-neg 99.9%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 94.8%

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

      11. sub-neg 94.8%

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

      12. difference-of-squares 94.8%

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

   8. Applied egg-rr 94.8%

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

   10. Applied egg-rr 94.8%

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

   if 0.309999999999999998 < x

   1. Initial program 4.9%

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

      1. flip-- 4.9%

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

      2. div-inv 4.9%

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

      3. add-sqr-sqrt 5.3%

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

      4. add-sqr-sqrt 5.9%

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

      5. associate--l+ 5.9%

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

   4. Applied egg-rr 5.9%

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

      1. associate-*r/ 5.9%

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

      2. *-rgt-identity 5.9%

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

      3. +-commutative 5.9%

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

      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%

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

   6. Simplified 99.6%

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

   8. Applied egg-rr 50.9%

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

      1. associate-*l/ 51.1%

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

      2. *-lft-identity 51.1%

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

      3. associate-+r+ 51.1%

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

      4. associate--l+ 51.1%

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

      5. count-2 51.1%

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

      6. distribute-lft1-in 51.1%

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

      7. unpow1 51.1%

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

      8. sqr-pow 51.1%

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

      9. hypot-def 68.3%

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

      10. metadata-eval 68.3%

          \[\leadsto \frac{2 \cdot x + \left(1 - \mathsf{hypot}\left(x, {x}^{\color{blue}{0.5}}\right)\right)}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \]

      11. unpow1/2 68.3%

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

      12. +-commutative 68.3%

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

   10. Simplified 68.3%

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

   11. Taylor expanded in x around inf 99.2%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.31:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, 1\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.31:\\ \;\;\;\;1 + x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.31) (+ 1.0 (* x -2.0)) (* 0.5 (sqrt (/ 1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.31) {
		tmp = 1.0 + (x * -2.0);
	} else {
		tmp = 0.5 * sqrt((1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.31d0) then
        tmp = 1.0d0 + (x * (-2.0d0))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.31) {
		tmp = 1.0 + (x * -2.0);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.31:
		tmp = 1.0 + (x * -2.0)
	else:
		tmp = 0.5 * math.sqrt((1.0 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.31)
		tmp = Float64(1.0 + Float64(x * -2.0));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.31)
		tmp = 1.0 + (x * -2.0);
	else
		tmp = 0.5 * sqrt((1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.31], N[(1.0 + N[(x * -2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.309999999999999998

   1. Initial program 99.9%

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

      1. flip-- 99.8%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l- 99.9%

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

      5. +-inverses 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 99.9%

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

      3. frac-times 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. sqr-neg 99.9%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 94.8%

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

      11. sub-neg 94.8%

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

      12. difference-of-squares 94.8%

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

   8. Applied egg-rr 94.8%

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

   10. Applied egg-rr 94.8%

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x, 2, 1\right)}} \]

   11. Taylor expanded in x around 0 94.8%

       \[\leadsto \color{blue}{1 + -2 \cdot x} \]

   if 0.309999999999999998 < x

   1. Initial program 4.9%

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

      1. flip-- 4.9%

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

      2. div-inv 4.9%

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

      3. add-sqr-sqrt 5.3%

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

      4. add-sqr-sqrt 5.9%

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

      5. associate--l+ 5.9%

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

   4. Applied egg-rr 5.9%

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

      1. associate-*r/ 5.9%

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

      2. *-rgt-identity 5.9%

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

      3. +-commutative 5.9%

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

      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%

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

   6. Simplified 99.6%

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

   8. Applied egg-rr 50.9%

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

      1. associate-*l/ 51.1%

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

      2. *-lft-identity 51.1%

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

      3. associate-+r+ 51.1%

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

      4. associate--l+ 51.1%

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

      5. count-2 51.1%

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

      6. distribute-lft1-in 51.1%

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

      7. unpow1 51.1%

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

      8. sqr-pow 51.1%

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

      9. hypot-def 68.3%

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

      10. metadata-eval 68.3%

          \[\leadsto \frac{2 \cdot x + \left(1 - \mathsf{hypot}\left(x, {x}^{\color{blue}{0.5}}\right)\right)}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \]

      11. unpow1/2 68.3%

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

      12. +-commutative 68.3%

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

   10. Simplified 68.3%

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

   11. Taylor expanded in x around inf 99.2%

       \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.31:\\ \;\;\;\;1 + x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.365:\\ \;\;\;\;1 + x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.365) (+ 1.0 (* x -2.0)) (sqrt (/ 0.5 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.365) {
		tmp = 1.0 + (x * -2.0);
	} else {
		tmp = sqrt((0.5 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.365d0) then
        tmp = 1.0d0 + (x * (-2.0d0))
    else
        tmp = sqrt((0.5d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.365) {
		tmp = 1.0 + (x * -2.0);
	} else {
		tmp = Math.sqrt((0.5 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.365:
		tmp = 1.0 + (x * -2.0)
	else:
		tmp = math.sqrt((0.5 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.365)
		tmp = Float64(1.0 + Float64(x * -2.0));
	else
		tmp = sqrt(Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.365)
		tmp = 1.0 + (x * -2.0);
	else
		tmp = sqrt((0.5 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.365], N[(1.0 + N[(x * -2.0), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.36499999999999999

   1. Initial program 99.9%

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

      1. flip-- 99.8%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+l- 99.9%

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

      5. +-inverses 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 99.9%

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

      3. frac-times 99.9%

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

      4. metadata-eval 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. sqr-neg 99.9%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 94.8%

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

      11. sub-neg 94.8%

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

      12. difference-of-squares 94.8%

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

   8. Applied egg-rr 94.8%

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

   10. Applied egg-rr 94.8%

       \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(x, 2, 1\right)}} \]

   11. Taylor expanded in x around 0 94.8%

       \[\leadsto \color{blue}{1 + -2 \cdot x} \]

   if 0.36499999999999999 < x

   1. Initial program 4.9%

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

      1. flip-- 4.9%

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

      2. div-inv 4.9%

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

      3. add-sqr-sqrt 5.3%

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

      4. add-sqr-sqrt 5.9%

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

      5. associate--l+ 5.9%

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

   4. Applied egg-rr 5.9%

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

      1. associate-*r/ 5.9%

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

      2. *-rgt-identity 5.9%

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

      3. +-commutative 5.9%

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

      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%

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

   6. Simplified 99.6%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 99.6%

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

      3. frac-times 98.9%

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

      4. metadata-eval 98.9%

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

      5. +-commutative 98.9%

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

      6. +-commutative 98.9%

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

      7. add-sqr-sqrt 98.9%

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

      8. sqr-neg 98.9%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 2.8%

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

      11. sub-neg 2.8%

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

      12. difference-of-squares 2.8%

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

   8. Applied egg-rr 20.3%

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

   9. Taylor expanded in x around inf 20.3%

      \[\leadsto \sqrt{\color{blue}{\frac{0.5}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.365:\\ \;\;\;\;1 + x \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.5}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 3.5% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 49.1%

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

   1. add-cbrt-cube 49.0%

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

   2. pow3 49.0%

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

4. Applied egg-rr 49.0%

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

   1. flip-- 49.0%

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

   2. div-inv 49.0%

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

   3. add-sqr-sqrt 49.3%

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

   4. add-sqr-sqrt 49.6%

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

   5. associate--l+ 49.6%

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

6. Applied egg-rr 49.6%

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

   1. associate-*r/ 49.6%

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

   2. *-rgt-identity 49.6%

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

   3. +-commutative 49.6%

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

   4. associate-+l- 99.7%

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

   5. +-inverses 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

8. Simplified 84.4%

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

9. Taylor expanded in x around 0 47.6%

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

11. Simplified 3.5%

    \[\leadsto \color{blue}{0} \]

12. Final simplification 3.5%

    \[\leadsto 0 \]
13. Add Preprocessing

Alternative 7: 13.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.6666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.6666666666666666)

C

double code(double x) {
	return 0.6666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.6666666666666666d0
end function

Java

public static double code(double x) {
	return 0.6666666666666666;
}

Python

def code(x):
	return 0.6666666666666666

Julia

function code(x)
	return 0.6666666666666666
end

MATLAB

function tmp = code(x)
	tmp = 0.6666666666666666;
end

Wolfram

code[x_] := 0.6666666666666666

Derivation

1. Initial program 49.1%

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

   1. flip-- 49.0%

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

   2. div-inv 49.0%

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

   3. add-sqr-sqrt 49.3%

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

   4. add-sqr-sqrt 49.6%

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

   5. associate--l+ 49.6%

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

4. Applied egg-rr 49.6%

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

   1. associate-*r/ 49.6%

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

   2. *-rgt-identity 49.6%

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

   3. +-commutative 49.6%

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

   4. associate-+l- 99.7%

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

   5. +-inverses 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

6. Simplified 99.7%

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

8. Applied egg-rr 73.7%

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

   1. associate-*l/ 73.8%

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

   2. *-lft-identity 73.8%

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

   3. associate-+r+ 73.8%

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

   4. associate--l+ 73.8%

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

   5. count-2 73.8%

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

   6. distribute-lft1-in 73.8%

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

   7. unpow1 73.8%

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

   8. sqr-pow 73.8%

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

   9. hypot-def 83.0%

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

   10. metadata-eval 83.0%

       \[\leadsto \frac{2 \cdot x + \left(1 - \mathsf{hypot}\left(x, {x}^{\color{blue}{0.5}}\right)\right)}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \]

   11. unpow1/2 83.0%

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

   12. +-commutative 83.0%

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

10. Simplified 83.0%

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

11. Taylor expanded in x around 0 53.3%

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

12. Taylor expanded in x around inf 12.7%

    \[\leadsto \color{blue}{0.6666666666666666} \]

13. Final simplification 12.7%

    \[\leadsto 0.6666666666666666 \]
14. Add Preprocessing

Alternative 8: 50.8% 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 49.1%

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

3. Taylor expanded in x around 0 47.6%

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

4. Final simplification 47.6%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 99.8% 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 2024019 
(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)))