2sqrt (example 3.1)

Percentage Accurate: 52.7% → 99.7%

Time: 5.7s

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: 52.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

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 52.6%

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

   1. flip-- 53.1%

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

   2. div-inv 53.1%

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

   3. add-sqr-sqrt 52.7%

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

   4. add-sqr-sqrt 53.9%

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

3. Applied egg-rr 53.9%

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

   1. *-commutative 53.9%

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

   2. associate-/r/ 53.9%

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

   3. +-commutative 53.9%

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

   4. associate--l+ 99.8%

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

   5. +-inverses 99.8%

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

   6. metadata-eval 99.8%

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

   7. /-rgt-identity 99.8%

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

   8. +-commutative 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = pow(x, -0.5) * 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 <= 5d-5) then
        tmp = (x ** (-0.5d0)) * 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 <= 5e-5) {
		tmp = Math.pow(x, -0.5) * 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 <= 5e-5:
		tmp = math.pow(x, -0.5) * 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 <= 5e-5)
		tmp = Float64((x ^ -0.5) * 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 <= 5e-5)
		tmp = (x ^ -0.5) * 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, 5e-5], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 5.00000000000000024e-5

   1. Initial program 5.2%

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

      1. flip3-- 3.4%

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

      2. div-inv 3.4%

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

      3. sqrt-pow2 3.7%

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

      4. metadata-eval 3.7%

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

      5. sqrt-pow2 3.6%

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

      6. metadata-eval 3.6%

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

      7. add-sqr-sqrt 3.6%

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

      8. add-sqr-sqrt 3.6%

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

      9. associate-+r+ 3.6%

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

      10. sqrt-unprod 3.6%

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

   3. Applied egg-rr 3.6%

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

   4. Taylor expanded in x around inf 99.2%

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

      1. *-commutative 99.2%

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

   6. Simplified 99.2%

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

      1. expm1-log1p-u 99.2%

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

      2. expm1-udef 9.2%

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

      3. sqrt-div 9.2%

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

      4. metadata-eval 9.2%

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

   8. Applied egg-rr 9.2%

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

      1. expm1-def 98.9%

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

      2. expm1-log1p 98.9%

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

   10. Simplified 98.9%

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

       1. add-log-exp 9.2%

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

       2. *-un-lft-identity 9.2%

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

       3. log-prod 9.2%

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

       4. metadata-eval 9.2%

          \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{\sqrt{x}}}\right)\right) \cdot 0.5 \]

       5. add-log-exp 98.9%

          \[\leadsto \left(0 + \color{blue}{\frac{1}{\sqrt{x}}}\right) \cdot 0.5 \]

       6. inv-pow 98.9%

          \[\leadsto \left(0 + \color{blue}{{\left(\sqrt{x}\right)}^{-1}}\right) \cdot 0.5 \]

       7. sqrt-pow2 99.3%

          \[\leadsto \left(0 + \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot 0.5 \]

       8. metadata-eval 99.3%

          \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) \cdot 0.5 \]

   12. Applied egg-rr 99.3%

       \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} \cdot 0.5 \]
   13. Step-by-step derivation

       1. +-lft-identity 99.3%

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

   14. Simplified 99.3%

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

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.3%

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

4. Final simplification 99.3%

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

Alternative 3: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{1}{x \cdot 0.5 + \left(1 + \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) (/ 1.0 (+ (* x 0.5) (+ 1.0 (sqrt x)))) (* (pow x -0.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0 / ((x * 0.5) + (1.0 + sqrt(x)));
	} else {
		tmp = pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = 1.0d0 / ((x * 0.5d0) + (1.0d0 + sqrt(x)))
    else
        tmp = (x ** (-0.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 1.0 / ((x * 0.5) + (1.0 + Math.sqrt(x)));
	} else {
		tmp = Math.pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = 1.0 / ((x * 0.5) + (1.0 + math.sqrt(x)))
	else:
		tmp = math.pow(x, -0.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = Float64(1.0 / Float64(Float64(x * 0.5) + Float64(1.0 + sqrt(x))));
	else
		tmp = Float64((x ^ -0.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = 1.0 / ((x * 0.5) + (1.0 + sqrt(x)));
	else
		tmp = (x ^ -0.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], N[(1.0 / N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 99.9%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 100.0%

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

      4. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--l+ 100.0%

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

      5. +-inverses 100.0%

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

      6. metadata-eval 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. +-commutative 100.0%

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

   5. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. fma-def 100.0%

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

      4. pow1/2 100.0%

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

      5. sqrt-pow1 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow1/2 100.0%

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

      8. sqrt-pow1 100.0%

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

      9. metadata-eval 100.0%

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

      10. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 98.7%

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

   if 2.39999999999999991 < x

   1. Initial program 7.4%

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

      1. flip3-- 5.7%

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

      2. div-inv 5.7%

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

      3. sqrt-pow2 5.9%

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

      4. metadata-eval 5.9%

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

      5. sqrt-pow2 5.9%

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

      6. metadata-eval 5.9%

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

      7. add-sqr-sqrt 5.9%

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

      8. add-sqr-sqrt 5.9%

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

      9. associate-+r+ 5.9%

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

      10. sqrt-unprod 5.9%

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

   3. Applied egg-rr 5.9%

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

   4. Taylor expanded in x around inf 97.5%

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

      1. *-commutative 97.5%

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

   6. Simplified 97.5%

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

      1. expm1-log1p-u 97.5%

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

      2. expm1-udef 10.3%

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

      3. sqrt-div 10.3%

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

      4. metadata-eval 10.3%

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

   8. Applied egg-rr 10.3%

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

      1. expm1-def 97.3%

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

      2. expm1-log1p 97.3%

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

   10. Simplified 97.3%

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

       1. add-log-exp 10.3%

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

       2. *-un-lft-identity 10.3%

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

       3. log-prod 10.3%

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

       4. metadata-eval 10.3%

          \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{\sqrt{x}}}\right)\right) \cdot 0.5 \]

       5. add-log-exp 97.3%

          \[\leadsto \left(0 + \color{blue}{\frac{1}{\sqrt{x}}}\right) \cdot 0.5 \]

       6. inv-pow 97.3%

          \[\leadsto \left(0 + \color{blue}{{\left(\sqrt{x}\right)}^{-1}}\right) \cdot 0.5 \]

       7. sqrt-pow2 97.7%

          \[\leadsto \left(0 + \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot 0.5 \]

       8. metadata-eval 97.7%

          \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) \cdot 0.5 \]

   12. Applied egg-rr 97.7%

       \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} \cdot 0.5 \]
   13. Step-by-step derivation

       1. +-lft-identity 97.7%

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

   14. Simplified 97.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\frac{1}{x \cdot 0.5 + \left(1 + \sqrt{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \]

Alternative 4: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = (1.0 - sqrt(x)) / (1.0 - x);
	} else {
		tmp = pow(x, -0.5) * 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 - sqrt(x)) / (1.0d0 - x)
    else
        tmp = (x ** (-0.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 100.0%

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

      4. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--l+ 100.0%

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

      5. +-inverses 100.0%

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

      6. metadata-eval 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. +-commutative 100.0%

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

   5. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. fma-def 100.0%

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

      4. pow1/2 100.0%

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

      5. sqrt-pow1 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow1/2 100.0%

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

      8. sqrt-pow1 100.0%

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

      9. metadata-eval 100.0%

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

      10. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 97.8%

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

      1. flip-+ 97.8%

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

      2. associate-/r/ 97.8%

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

      3. metadata-eval 97.8%

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

      4. add-sqr-sqrt 97.8%

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

   10. Applied egg-rr 97.8%

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

       1. associate-*l/ 97.8%

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

       2. *-lft-identity 97.8%

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

   12. Simplified 97.8%

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

   if 1 < x

   1. Initial program 7.4%

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

      1. flip3-- 5.7%

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

      2. div-inv 5.7%

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

      3. sqrt-pow2 5.9%

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

      4. metadata-eval 5.9%

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

      5. sqrt-pow2 5.9%

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

      6. metadata-eval 5.9%

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

      7. add-sqr-sqrt 5.9%

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

      8. add-sqr-sqrt 5.9%

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

      9. associate-+r+ 5.9%

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

      10. sqrt-unprod 5.9%

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

   3. Applied egg-rr 5.9%

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

   4. Taylor expanded in x around inf 97.5%

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

      1. *-commutative 97.5%

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

   6. Simplified 97.5%

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

      1. expm1-log1p-u 97.5%

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

      2. expm1-udef 10.3%

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

      3. sqrt-div 10.3%

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

      4. metadata-eval 10.3%

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

   8. Applied egg-rr 10.3%

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

      1. expm1-def 97.3%

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

      2. expm1-log1p 97.3%

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

   10. Simplified 97.3%

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

       1. add-log-exp 10.3%

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

       2. *-un-lft-identity 10.3%

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

       3. log-prod 10.3%

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

       4. metadata-eval 10.3%

          \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{\sqrt{x}}}\right)\right) \cdot 0.5 \]

       5. add-log-exp 97.3%

          \[\leadsto \left(0 + \color{blue}{\frac{1}{\sqrt{x}}}\right) \cdot 0.5 \]

       6. inv-pow 97.3%

          \[\leadsto \left(0 + \color{blue}{{\left(\sqrt{x}\right)}^{-1}}\right) \cdot 0.5 \]

       7. sqrt-pow2 97.7%

          \[\leadsto \left(0 + \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot 0.5 \]

       8. metadata-eval 97.7%

          \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) \cdot 0.5 \]

   12. Applied egg-rr 97.7%

       \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} \cdot 0.5 \]
   13. Step-by-step derivation

       1. +-lft-identity 97.7%

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

   14. Simplified 97.7%

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

4. Final simplification 97.7%

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

Alternative 5: 98.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 / (1.0 + math.sqrt(x))
	else:
		tmp = math.pow(x, -0.5) * 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((x ^ -0.5) * 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 = (x ^ -0.5) * 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[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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 100.0%

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

      4. add-sqr-sqrt 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 100.0%

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

      3. +-commutative 100.0%

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

      4. associate--l+ 100.0%

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

      5. +-inverses 100.0%

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

      6. metadata-eval 100.0%

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

      7. /-rgt-identity 100.0%

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

      8. +-commutative 100.0%

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

   5. Simplified 100.0%

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

      1. +-commutative 100.0%

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

      2. add-sqr-sqrt 100.0%

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

      3. fma-def 100.0%

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

      4. pow1/2 100.0%

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

      5. sqrt-pow1 100.0%

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

      6. metadata-eval 100.0%

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

      7. pow1/2 100.0%

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

      8. sqrt-pow1 100.0%

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

      9. metadata-eval 100.0%

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

      10. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in x around 0 97.8%

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

   if 1 < x

   1. Initial program 7.4%

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

      1. flip3-- 5.7%

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

      2. div-inv 5.7%

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

      3. sqrt-pow2 5.9%

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

      4. metadata-eval 5.9%

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

      5. sqrt-pow2 5.9%

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

      6. metadata-eval 5.9%

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

      7. add-sqr-sqrt 5.9%

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

      8. add-sqr-sqrt 5.9%

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

      9. associate-+r+ 5.9%

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

      10. sqrt-unprod 5.9%

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

   3. Applied egg-rr 5.9%

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

   4. Taylor expanded in x around inf 97.5%

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

      1. *-commutative 97.5%

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

   6. Simplified 97.5%

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

      1. expm1-log1p-u 97.5%

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

      2. expm1-udef 10.3%

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

      3. sqrt-div 10.3%

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

      4. metadata-eval 10.3%

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

   8. Applied egg-rr 10.3%

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

      1. expm1-def 97.3%

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

      2. expm1-log1p 97.3%

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

   10. Simplified 97.3%

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

       1. add-log-exp 10.3%

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

       2. *-un-lft-identity 10.3%

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

       3. log-prod 10.3%

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

       4. metadata-eval 10.3%

          \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{\sqrt{x}}}\right)\right) \cdot 0.5 \]

       5. add-log-exp 97.3%

          \[\leadsto \left(0 + \color{blue}{\frac{1}{\sqrt{x}}}\right) \cdot 0.5 \]

       6. inv-pow 97.3%

          \[\leadsto \left(0 + \color{blue}{{\left(\sqrt{x}\right)}^{-1}}\right) \cdot 0.5 \]

       7. sqrt-pow2 97.7%

          \[\leadsto \left(0 + \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot 0.5 \]

       8. metadata-eval 97.7%

          \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) \cdot 0.5 \]

   12. Applied egg-rr 97.7%

       \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} \cdot 0.5 \]
   13. Step-by-step derivation

       1. +-lft-identity 97.7%

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

   14. Simplified 97.7%

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

4. Final simplification 97.7%

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

Alternative 6: 96.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 96.1%

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

   if 0.25 < x

   1. Initial program 7.4%

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

      1. flip3-- 5.7%

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

      2. div-inv 5.7%

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

      3. sqrt-pow2 5.9%

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

      4. metadata-eval 5.9%

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

      5. sqrt-pow2 5.9%

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

      6. metadata-eval 5.9%

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

      7. add-sqr-sqrt 5.9%

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

      8. add-sqr-sqrt 5.9%

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

      9. associate-+r+ 5.9%

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

      10. sqrt-unprod 5.9%

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

   3. Applied egg-rr 5.9%

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

   4. Taylor expanded in x around inf 97.5%

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

      1. *-commutative 97.5%

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

   6. Simplified 97.5%

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

      1. expm1-log1p-u 97.5%

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

      2. expm1-udef 10.3%

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

      3. sqrt-div 10.3%

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

      4. metadata-eval 10.3%

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

   8. Applied egg-rr 10.3%

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

      1. expm1-def 97.3%

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

      2. expm1-log1p 97.3%

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

   10. Simplified 97.3%

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

       1. add-log-exp 10.3%

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

       2. *-un-lft-identity 10.3%

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

       3. log-prod 10.3%

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

       4. metadata-eval 10.3%

          \[\leadsto \left(\color{blue}{0} + \log \left(e^{\frac{1}{\sqrt{x}}}\right)\right) \cdot 0.5 \]

       5. add-log-exp 97.3%

          \[\leadsto \left(0 + \color{blue}{\frac{1}{\sqrt{x}}}\right) \cdot 0.5 \]

       6. inv-pow 97.3%

          \[\leadsto \left(0 + \color{blue}{{\left(\sqrt{x}\right)}^{-1}}\right) \cdot 0.5 \]

       7. sqrt-pow2 97.7%

          \[\leadsto \left(0 + \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right) \cdot 0.5 \]

       8. metadata-eval 97.7%

          \[\leadsto \left(0 + {x}^{\color{blue}{-0.5}}\right) \cdot 0.5 \]

   12. Applied egg-rr 97.7%

       \[\leadsto \color{blue}{\left(0 + {x}^{-0.5}\right)} \cdot 0.5 \]
   13. Step-by-step derivation

       1. +-lft-identity 97.7%

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

   14. Simplified 97.7%

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

4. Final simplification 96.9%

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

Alternative 7: 96.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.25:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.25) 1.0 (/ 0.5 (sqrt x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = 0.5 / sqrt(x);
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.25) {
		tmp = 1.0;
	} else {
		tmp = 0.5 / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.25:
		tmp = 1.0
	else:
		tmp = 0.5 / math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = Float64(0.5 / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.25)
		tmp = 1.0;
	else
		tmp = 0.5 / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.25], 1.0, N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 96.1%

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

   if 0.25 < x

   1. Initial program 7.4%

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

      1. flip3-- 5.7%

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

      2. div-inv 5.7%

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

      3. sqrt-pow2 5.9%

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

      4. metadata-eval 5.9%

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

      5. sqrt-pow2 5.9%

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

      6. metadata-eval 5.9%

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

      7. add-sqr-sqrt 5.9%

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

      8. add-sqr-sqrt 5.9%

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

      9. associate-+r+ 5.9%

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

      10. sqrt-unprod 5.9%

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

   3. Applied egg-rr 5.9%

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

   4. Taylor expanded in x around inf 48.3%

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

   5. Taylor expanded in x around inf 97.0%

      \[\leadsto \left(1.5 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{0.3333333333333333}{x}} \]
   6. Step-by-step derivation

      1. add-log-exp 9.7%

         \[\leadsto \color{blue}{\log \left(e^{\left(1.5 \cdot \sqrt{x}\right) \cdot \frac{0.3333333333333333}{x}}\right)} \]

      2. *-un-lft-identity 9.7%

         \[\leadsto \log \color{blue}{\left(1 \cdot e^{\left(1.5 \cdot \sqrt{x}\right) \cdot \frac{0.3333333333333333}{x}}\right)} \]

      3. log-prod 9.7%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\left(1.5 \cdot \sqrt{x}\right) \cdot \frac{0.3333333333333333}{x}}\right)} \]

      4. metadata-eval 9.7%

         \[\leadsto \color{blue}{0} + \log \left(e^{\left(1.5 \cdot \sqrt{x}\right) \cdot \frac{0.3333333333333333}{x}}\right) \]

      5. add-log-exp 97.0%

         \[\leadsto 0 + \color{blue}{\left(1.5 \cdot \sqrt{x}\right) \cdot \frac{0.3333333333333333}{x}} \]

      6. associate-*r/ 96.9%

         \[\leadsto 0 + \color{blue}{\frac{\left(1.5 \cdot \sqrt{x}\right) \cdot 0.3333333333333333}{x}} \]

      7. *-commutative 96.9%

         \[\leadsto 0 + \frac{\color{blue}{\left(\sqrt{x} \cdot 1.5\right)} \cdot 0.3333333333333333}{x} \]

      8. associate-*l* 97.3%

         \[\leadsto 0 + \frac{\color{blue}{\sqrt{x} \cdot \left(1.5 \cdot 0.3333333333333333\right)}}{x} \]

      9. metadata-eval 97.3%

         \[\leadsto 0 + \frac{\sqrt{x} \cdot \color{blue}{0.5}}{x} \]

   7. Applied egg-rr 97.3%

      \[\leadsto \color{blue}{0 + \frac{\sqrt{x} \cdot 0.5}{x}} \]
   8. Step-by-step derivation

      1. +-lft-identity 97.3%

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

      2. rem-square-sqrt 97.1%

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

      3. associate-/l/ 97.3%

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

      4. associate-/l* 97.3%

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

      5. associate-/r/ 97.3%

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

      6. *-inverses 97.3%

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

      7. metadata-eval 97.3%

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

   9. Simplified 97.3%

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

4. Final simplification 96.7%

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

Alternative 8: 50.7% 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 52.6%

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

2. Taylor expanded in x around 0 50.4%

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

3. Final simplification 50.4%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023192 
(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)))