2sqrt (example 3.1)

Percentage Accurate: 53.9% → 99.8%

Time: 6.4s

Alternatives: 7

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.8%

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

   1. flip-- 53.3%

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

   2. div-inv 53.3%

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

   3. add-sqr-sqrt 53.6%

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

   4. add-sqr-sqrt 54.1%

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

3. Applied egg-rr 54.1%

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

   1. associate-*r/ 54.1%

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

   2. *-rgt-identity 54.1%

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

   3. remove-double-neg 54.1%

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

   4. sub-neg 54.1%

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

   5. div-sub 52.7%

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

   6. rem-square-sqrt 52.6%

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

   7. sqr-neg 52.6%

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

   8. div-sub 53.6%

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

   9. sqr-neg 53.6%

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

   10. +-commutative 53.6%

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

   11. rem-square-sqrt 54.1%

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

   12. associate--l+ 99.7%

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

   13. +-inverses 99.7%

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

   14. metadata-eval 99.7%

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

   15. sub-neg 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 0.0001:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 0.0001) (* 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 <= 0.0001) {
		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 <= 0.0001d0) 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 <= 0.0001) {
		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 <= 0.0001:
		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 <= 0.0001)
		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 <= 0.0001)
		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, 0.0001], 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)) < 1.00000000000000005e-4

   1. Initial program 6.0%

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

      1. flip-- 7.2%

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

      2. div-inv 7.2%

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

      3. add-sqr-sqrt 7.5%

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

      4. add-sqr-sqrt 8.4%

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

   3. Applied egg-rr 8.4%

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

      1. associate-*r/ 8.4%

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

      2. *-rgt-identity 8.4%

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

      3. remove-double-neg 8.4%

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

      4. sub-neg 8.4%

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

      5. div-sub 6.0%

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

      6. rem-square-sqrt 5.7%

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

      7. sqr-neg 5.7%

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

      8. div-sub 7.5%

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

      9. sqr-neg 7.5%

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

      10. +-commutative 7.5%

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

      11. rem-square-sqrt 8.4%

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

      12. associate--l+ 99.6%

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

      13. +-inverses 99.6%

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

      14. metadata-eval 99.6%

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

      15. sub-neg 99.6%

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

   5. Simplified 99.6%

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

      1. flip3-+ 56.3%

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

      2. associate-/r/ 56.3%

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

      3. sqrt-pow2 56.3%

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

      4. +-commutative 56.3%

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

      5. metadata-eval 56.3%

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

      6. sqrt-pow2 56.1%

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

      7. metadata-eval 56.1%

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

      8. add-sqr-sqrt 56.4%

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

      9. add-sqr-sqrt 56.1%

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

      10. associate-+r- 56.1%

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

   7. Applied egg-rr 39.8%

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

   8. Taylor expanded in x around inf 56.4%

      \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(0.5 + \left(0.125 \cdot \frac{1}{x} + x\right)\right) - 0.0625 \cdot \frac{1}{{x}^{2}}\right)} \]
   9. Step-by-step derivation

      1. sub-neg 56.4%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(0.5 + \left(0.125 \cdot \frac{1}{x} + x\right)\right) + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      2. +-commutative 56.4%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(\left(0.125 \cdot \frac{1}{x} + x\right) + 0.5\right)} + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) \]

      3. associate-+l+ 56.4%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(0.125 \cdot \frac{1}{x} + x\right) + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      4. +-commutative 56.4%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(x + 0.125 \cdot \frac{1}{x}\right)} + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ 56.4%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \color{blue}{\frac{0.125 \cdot 1}{x}}\right) + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval 56.4%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{\color{blue}{0.125}}{x}\right) + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      7. associate-*r/ 56.4%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \left(-\color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right)\right)\right) \]

      8. metadata-eval 56.4%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \left(-\frac{\color{blue}{0.0625}}{{x}^{2}}\right)\right)\right) \]

      9. distribute-neg-frac 56.4%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \color{blue}{\frac{-0.0625}{{x}^{2}}}\right)\right) \]

      10. metadata-eval 56.4%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \frac{\color{blue}{-0.0625}}{{x}^{2}}\right)\right) \]

      11. unpow2 56.4%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \frac{-0.0625}{\color{blue}{x \cdot x}}\right)\right) \]

   10. Simplified 56.4%

       \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \frac{-0.0625}{x \cdot x}\right)\right)} \]

   11. Taylor expanded in x around inf 98.8%

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

       1. unpow1/2 98.8%

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

       2. rem-exp-log 91.4%

          \[\leadsto 0.5 \cdot {\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \]

       3. exp-neg 91.4%

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

       4. exp-prod 91.3%

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

       5. distribute-lft-neg-out 91.3%

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

       6. distribute-rgt-neg-in 91.3%

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

       7. metadata-eval 91.3%

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

       8. exp-to-pow 98.9%

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

   13. Simplified 98.9%

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

   if 1.00000000000000005e-4 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0.0001:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.3%

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

   if 1 < x

   1. Initial program 7.7%

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

      1. flip-- 8.9%

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

      2. div-inv 8.9%

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

      3. add-sqr-sqrt 9.4%

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

      4. add-sqr-sqrt 10.5%

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

   3. Applied egg-rr 10.5%

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

      1. associate-*r/ 10.5%

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

      2. *-rgt-identity 10.5%

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

      3. remove-double-neg 10.5%

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

      4. sub-neg 10.5%

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

      5. div-sub 7.8%

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

      6. rem-square-sqrt 7.6%

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

      7. sqr-neg 7.6%

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

      8. div-sub 9.4%

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

      9. sqr-neg 9.4%

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

      10. +-commutative 9.4%

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

      11. rem-square-sqrt 10.5%

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

      12. associate--l+ 99.6%

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

      13. +-inverses 99.6%

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

      14. metadata-eval 99.6%

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

      15. sub-neg 99.6%

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

   5. Simplified 99.6%

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

      1. flip3-+ 57.3%

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

      2. associate-/r/ 57.3%

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

      3. sqrt-pow2 57.3%

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

      4. +-commutative 57.3%

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

      5. metadata-eval 57.3%

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

      6. sqrt-pow2 57.1%

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

      7. metadata-eval 57.1%

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

      8. add-sqr-sqrt 57.3%

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

      9. add-sqr-sqrt 57.1%

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

      10. associate-+r- 57.1%

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

   7. Applied egg-rr 41.1%

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

   8. Taylor expanded in x around inf 57.3%

      \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(0.5 + \left(0.125 \cdot \frac{1}{x} + x\right)\right) - 0.0625 \cdot \frac{1}{{x}^{2}}\right)} \]
   9. Step-by-step derivation

      1. sub-neg 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(0.5 + \left(0.125 \cdot \frac{1}{x} + x\right)\right) + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      2. +-commutative 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(\left(0.125 \cdot \frac{1}{x} + x\right) + 0.5\right)} + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) \]

      3. associate-+l+ 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(0.125 \cdot \frac{1}{x} + x\right) + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      4. +-commutative 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(x + 0.125 \cdot \frac{1}{x}\right)} + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \color{blue}{\frac{0.125 \cdot 1}{x}}\right) + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{\color{blue}{0.125}}{x}\right) + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      7. associate-*r/ 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \left(-\color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right)\right)\right) \]

      8. metadata-eval 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \left(-\frac{\color{blue}{0.0625}}{{x}^{2}}\right)\right)\right) \]

      9. distribute-neg-frac 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \color{blue}{\frac{-0.0625}{{x}^{2}}}\right)\right) \]

      10. metadata-eval 57.3%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \frac{\color{blue}{-0.0625}}{{x}^{2}}\right)\right) \]

      11. unpow2 57.3%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \frac{-0.0625}{\color{blue}{x \cdot x}}\right)\right) \]

   10. Simplified 57.3%

       \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \frac{-0.0625}{x \cdot x}\right)\right)} \]

   11. Taylor expanded in x around inf 97.4%

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

       1. unpow1/2 97.4%

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

       2. rem-exp-log 90.2%

          \[\leadsto 0.5 \cdot {\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \]

       3. exp-neg 90.2%

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

       4. exp-prod 90.2%

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

       5. distribute-lft-neg-out 90.2%

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

       6. distribute-rgt-neg-in 90.2%

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

       7. metadata-eval 90.2%

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

       8. exp-to-pow 97.6%

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

   13. Simplified 97.6%

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

4. Final simplification 98.4%

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

Alternative 4: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 100.0%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 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%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. div-sub 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. sqr-neg 99.9%

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

      8. div-sub 99.9%

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

      9. sqr-neg 99.9%

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

      10. +-commutative 99.9%

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

      11. rem-square-sqrt 99.9%

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

      12. associate--l+ 99.9%

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

      13. +-inverses 99.9%

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

      14. metadata-eval 99.9%

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

      15. sub-neg 99.9%

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

   5. Simplified 99.9%

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

      1. +-commutative 99.9%

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

      2. add-sqr-sqrt 99.9%

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

      3. fma-def 99.9%

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

      4. pow1/2 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\sqrt{\color{blue}{{x}^{0.5}}}, \sqrt{\sqrt{x}}, \sqrt{1 + x}\right)} \]

      5. sqrt-pow1 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{x}}, \sqrt{1 + x}\right)} \]

      6. metadata-eval 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left({x}^{\color{blue}{0.25}}, \sqrt{\sqrt{x}}, \sqrt{1 + x}\right)} \]

      7. pow1/2 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left({x}^{0.25}, \sqrt{\color{blue}{{x}^{0.5}}}, \sqrt{1 + x}\right)} \]

      8. sqrt-pow1 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left({x}^{0.25}, \color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}, \sqrt{1 + x}\right)} \]

      9. metadata-eval 99.9%

         \[\leadsto \frac{1}{\mathsf{fma}\left({x}^{0.25}, {x}^{\color{blue}{0.25}}, \sqrt{1 + x}\right)} \]

      10. +-commutative 99.9%

          \[\leadsto \frac{1}{\mathsf{fma}\left({x}^{0.25}, {x}^{0.25}, \sqrt{\color{blue}{x + 1}}\right)} \]

   7. Applied egg-rr 99.9%

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left({x}^{0.25}, {x}^{0.25}, \sqrt{x + 1}\right)}} \]

   8. Taylor expanded in x around 0 98.9%

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

   if 1 < x

   1. Initial program 7.7%

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

      1. flip-- 8.9%

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

      2. div-inv 8.9%

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

      3. add-sqr-sqrt 9.4%

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

      4. add-sqr-sqrt 10.5%

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

   3. Applied egg-rr 10.5%

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

      1. associate-*r/ 10.5%

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

      2. *-rgt-identity 10.5%

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

      3. remove-double-neg 10.5%

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

      4. sub-neg 10.5%

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

      5. div-sub 7.8%

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

      6. rem-square-sqrt 7.6%

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

      7. sqr-neg 7.6%

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

      8. div-sub 9.4%

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

      9. sqr-neg 9.4%

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

      10. +-commutative 9.4%

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

      11. rem-square-sqrt 10.5%

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

      12. associate--l+ 99.6%

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

      13. +-inverses 99.6%

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

      14. metadata-eval 99.6%

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

      15. sub-neg 99.6%

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

   5. Simplified 99.6%

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

      1. flip3-+ 57.3%

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

      2. associate-/r/ 57.3%

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

      3. sqrt-pow2 57.3%

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

      4. +-commutative 57.3%

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

      5. metadata-eval 57.3%

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

      6. sqrt-pow2 57.1%

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

      7. metadata-eval 57.1%

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

      8. add-sqr-sqrt 57.3%

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

      9. add-sqr-sqrt 57.1%

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

      10. associate-+r- 57.1%

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

   7. Applied egg-rr 41.1%

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

   8. Taylor expanded in x around inf 57.3%

      \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(0.5 + \left(0.125 \cdot \frac{1}{x} + x\right)\right) - 0.0625 \cdot \frac{1}{{x}^{2}}\right)} \]
   9. Step-by-step derivation

      1. sub-neg 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(0.5 + \left(0.125 \cdot \frac{1}{x} + x\right)\right) + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      2. +-commutative 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(\left(0.125 \cdot \frac{1}{x} + x\right) + 0.5\right)} + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) \]

      3. associate-+l+ 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(0.125 \cdot \frac{1}{x} + x\right) + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      4. +-commutative 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(x + 0.125 \cdot \frac{1}{x}\right)} + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \color{blue}{\frac{0.125 \cdot 1}{x}}\right) + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{\color{blue}{0.125}}{x}\right) + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      7. associate-*r/ 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \left(-\color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right)\right)\right) \]

      8. metadata-eval 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \left(-\frac{\color{blue}{0.0625}}{{x}^{2}}\right)\right)\right) \]

      9. distribute-neg-frac 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \color{blue}{\frac{-0.0625}{{x}^{2}}}\right)\right) \]

      10. metadata-eval 57.3%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \frac{\color{blue}{-0.0625}}{{x}^{2}}\right)\right) \]

      11. unpow2 57.3%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \frac{-0.0625}{\color{blue}{x \cdot x}}\right)\right) \]

   10. Simplified 57.3%

       \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \frac{-0.0625}{x \cdot x}\right)\right)} \]

   11. Taylor expanded in x around inf 97.4%

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

       1. unpow1/2 97.4%

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

       2. rem-exp-log 90.2%

          \[\leadsto 0.5 \cdot {\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \]

       3. exp-neg 90.2%

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

       4. exp-prod 90.2%

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

       5. distribute-lft-neg-out 90.2%

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

       6. distribute-rgt-neg-in 90.2%

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

       7. metadata-eval 90.2%

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

       8. exp-to-pow 97.6%

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

   13. Simplified 97.6%

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

4. Final simplification 98.2%

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

Alternative 5: 96.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.58) (- 1.0 x) (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 0.58) {
		tmp = 1.0 - x;
	} else {
		tmp = 0.5 * pow(x, -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.58d0) then
        tmp = 1.0d0 - x
    else
        tmp = 0.5d0 * (x ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.58) {
		tmp = 1.0 - x;
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.58:
		tmp = 1.0 - x
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.58)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(0.5 * (x ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.58)
		tmp = 1.0 - x;
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.58], N[(1.0 - x), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.57999999999999996

   1. Initial program 100.0%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 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%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. div-sub 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. sqr-neg 99.9%

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

      8. div-sub 99.9%

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

      9. sqr-neg 99.9%

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

      10. +-commutative 99.9%

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

      11. rem-square-sqrt 99.9%

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

      12. associate--l+ 99.9%

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

      13. +-inverses 99.9%

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

      14. metadata-eval 99.9%

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

      15. sub-neg 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
   6. 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. clear-num 99.9%

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

      4. frac-times 99.9%

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

      5. metadata-eval 99.9%

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

      6. /-rgt-identity 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. sqr-neg 99.9%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 97.2%

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

      11. sub-neg 97.2%

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

      12. difference-of-squares 97.2%

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

      13. add-sqr-sqrt 97.2%

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

      14. +-commutative 97.2%

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

      15. add-sqr-sqrt 97.2%

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

      16. associate--l+ 97.2%

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

   7. Applied egg-rr 97.2%

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

   8. Taylor expanded in x around 0 97.2%

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

      1. mul-1-neg 97.2%

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

      2. sub-neg 97.2%

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

   10. Simplified 97.2%

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

   if 0.57999999999999996 < x

   1. Initial program 7.7%

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

      1. flip-- 8.9%

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

      2. div-inv 8.9%

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

      3. add-sqr-sqrt 9.4%

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

      4. add-sqr-sqrt 10.5%

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

   3. Applied egg-rr 10.5%

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

      1. associate-*r/ 10.5%

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

      2. *-rgt-identity 10.5%

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

      3. remove-double-neg 10.5%

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

      4. sub-neg 10.5%

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

      5. div-sub 7.8%

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

      6. rem-square-sqrt 7.6%

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

      7. sqr-neg 7.6%

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

      8. div-sub 9.4%

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

      9. sqr-neg 9.4%

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

      10. +-commutative 9.4%

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

      11. rem-square-sqrt 10.5%

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

      12. associate--l+ 99.6%

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

      13. +-inverses 99.6%

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

      14. metadata-eval 99.6%

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

      15. sub-neg 99.6%

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

   5. Simplified 99.6%

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

      1. flip3-+ 57.3%

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

      2. associate-/r/ 57.3%

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

      3. sqrt-pow2 57.3%

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

      4. +-commutative 57.3%

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

      5. metadata-eval 57.3%

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

      6. sqrt-pow2 57.1%

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

      7. metadata-eval 57.1%

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

      8. add-sqr-sqrt 57.3%

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

      9. add-sqr-sqrt 57.1%

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

      10. associate-+r- 57.1%

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

   7. Applied egg-rr 41.1%

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

   8. Taylor expanded in x around inf 57.3%

      \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(0.5 + \left(0.125 \cdot \frac{1}{x} + x\right)\right) - 0.0625 \cdot \frac{1}{{x}^{2}}\right)} \]
   9. Step-by-step derivation

      1. sub-neg 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(0.5 + \left(0.125 \cdot \frac{1}{x} + x\right)\right) + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)} \]

      2. +-commutative 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(\left(0.125 \cdot \frac{1}{x} + x\right) + 0.5\right)} + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right) \]

      3. associate-+l+ 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(0.125 \cdot \frac{1}{x} + x\right) + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right)} \]

      4. +-commutative 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\color{blue}{\left(x + 0.125 \cdot \frac{1}{x}\right)} + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      5. associate-*r/ 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \color{blue}{\frac{0.125 \cdot 1}{x}}\right) + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      6. metadata-eval 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{\color{blue}{0.125}}{x}\right) + \left(0.5 + \left(-0.0625 \cdot \frac{1}{{x}^{2}}\right)\right)\right) \]

      7. associate-*r/ 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \left(-\color{blue}{\frac{0.0625 \cdot 1}{{x}^{2}}}\right)\right)\right) \]

      8. metadata-eval 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \left(-\frac{\color{blue}{0.0625}}{{x}^{2}}\right)\right)\right) \]

      9. distribute-neg-frac 57.3%

         \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \color{blue}{\frac{-0.0625}{{x}^{2}}}\right)\right) \]

      10. metadata-eval 57.3%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \frac{\color{blue}{-0.0625}}{{x}^{2}}\right)\right) \]

      11. unpow2 57.3%

          \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \frac{-0.0625}{\color{blue}{x \cdot x}}\right)\right) \]

   10. Simplified 57.3%

       \[\leadsto \frac{1}{{\left(x + 1\right)}^{1.5} + {x}^{1.5}} \cdot \color{blue}{\left(\left(x + \frac{0.125}{x}\right) + \left(0.5 + \frac{-0.0625}{x \cdot x}\right)\right)} \]

   11. Taylor expanded in x around inf 97.4%

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

       1. unpow1/2 97.4%

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

       2. rem-exp-log 90.2%

          \[\leadsto 0.5 \cdot {\left(\frac{1}{\color{blue}{e^{\log x}}}\right)}^{0.5} \]

       3. exp-neg 90.2%

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

       4. exp-prod 90.2%

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

       5. distribute-lft-neg-out 90.2%

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

       6. distribute-rgt-neg-in 90.2%

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

       7. metadata-eval 90.2%

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

       8. exp-to-pow 97.6%

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

   13. Simplified 97.6%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.58:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]

Alternative 6: 58.5% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.74) (- 1.0 x) (sqrt (/ 0.5 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.74) {
		tmp = 1.0 - x;
	} else {
		tmp = sqrt((0.5 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.74d0) then
        tmp = 1.0d0 - x
    else
        tmp = sqrt((0.5d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.74) {
		tmp = 1.0 - x;
	} else {
		tmp = Math.sqrt((0.5 / x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.74:
		tmp = 1.0 - x
	else:
		tmp = math.sqrt((0.5 / x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.74)
		tmp = Float64(1.0 - x);
	else
		tmp = sqrt(Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.74)
		tmp = 1.0 - x;
	else
		tmp = sqrt((0.5 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.74], N[(1.0 - x), $MachinePrecision], N[Sqrt[N[(0.5 / x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.73999999999999999

   1. Initial program 100.0%

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

      1. flip-- 99.9%

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

      2. div-inv 99.9%

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

      3. add-sqr-sqrt 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%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 99.9%

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

      2. *-rgt-identity 99.9%

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

      3. remove-double-neg 99.9%

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

      4. sub-neg 99.9%

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

      5. div-sub 99.9%

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

      6. rem-square-sqrt 99.9%

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

      7. sqr-neg 99.9%

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

      8. div-sub 99.9%

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

      9. sqr-neg 99.9%

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

      10. +-commutative 99.9%

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

      11. rem-square-sqrt 99.9%

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

      12. associate--l+ 99.9%

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

      13. +-inverses 99.9%

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

      14. metadata-eval 99.9%

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

      15. sub-neg 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
   6. 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. clear-num 99.9%

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

      4. frac-times 99.9%

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

      5. metadata-eval 99.9%

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

      6. /-rgt-identity 99.9%

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

      7. add-sqr-sqrt 99.9%

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

      8. sqr-neg 99.9%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 97.2%

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

      11. sub-neg 97.2%

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

      12. difference-of-squares 97.2%

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

      13. add-sqr-sqrt 97.2%

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

      14. +-commutative 97.2%

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

      15. add-sqr-sqrt 97.2%

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

      16. associate--l+ 97.2%

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

   7. Applied egg-rr 97.2%

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

   8. Taylor expanded in x around 0 97.2%

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

      1. mul-1-neg 97.2%

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

      2. sub-neg 97.2%

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

   10. Simplified 97.2%

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

   if 0.73999999999999999 < x

   1. Initial program 7.7%

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

      1. flip-- 8.9%

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

      2. div-inv 8.9%

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

      3. add-sqr-sqrt 9.4%

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

      4. add-sqr-sqrt 10.5%

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

   3. Applied egg-rr 10.5%

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

      1. associate-*r/ 10.5%

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

      2. *-rgt-identity 10.5%

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

      3. remove-double-neg 10.5%

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

      4. sub-neg 10.5%

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

      5. div-sub 7.8%

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

      6. rem-square-sqrt 7.6%

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

      7. sqr-neg 7.6%

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

      8. div-sub 9.4%

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

      9. sqr-neg 9.4%

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

      10. +-commutative 9.4%

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

      11. rem-square-sqrt 10.5%

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

      12. associate--l+ 99.6%

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

      13. +-inverses 99.6%

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

      14. metadata-eval 99.6%

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

      15. sub-neg 99.6%

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1 + x} + \sqrt{x}}} \]
   6. 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 99.5%

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

      3. clear-num 99.5%

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

      4. frac-times 99.4%

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

      5. metadata-eval 99.4%

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

      6. /-rgt-identity 99.4%

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

      7. add-sqr-sqrt 99.4%

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

      8. sqr-neg 99.4%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 3.3%

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

      11. sub-neg 3.3%

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

      12. difference-of-squares 3.3%

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

      13. add-sqr-sqrt 4.9%

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

      14. +-commutative 4.9%

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

      15. add-sqr-sqrt 3.3%

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

      16. associate--l+ 3.3%

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

   7. Applied egg-rr 20.3%

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

   8. 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 57.9%

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

Alternative 7: 51.8% 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 52.8%

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

2. Taylor expanded in x around 0 50.9%

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

3. Final simplification 50.9%

   \[\leadsto 1 \]

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 2023175 
(FPCore (x)
  :name "2sqrt (example 3.1)"
  :precision binary64

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

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