Main:bigenough3 from C

Percentage Accurate: 54.5% → 99.0%

Time: 9.1s

Alternatives: 10

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: 54.5% 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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 0.0) (* 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 <= 0.0) {
		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 <= 0.0d0) 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 <= 0.0) {
		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 <= 0.0:
		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 <= 0.0)
		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 <= 0.0)
		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, 0.0], 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)) < 0.0

   1. Initial program 3.9%

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

      1. flip-- 3.9%

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

      2. div-inv 3.9%

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

      3. add-sqr-sqrt 3.6%

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

      4. add-sqr-sqrt 3.9%

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

      5. associate--l+ 3.9%

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

   4. Applied egg-rr 3.9%

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

      1. +-commutative 3.9%

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

      2. associate-+l- 99.5%

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

      3. +-inverses 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate-*r/ 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. inv-pow 99.5%

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

      2. add-sqr-sqrt 99.2%

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

      3. unpow-prod-down 99.1%

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

      4. add-sqr-sqrt 99.0%

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

      5. add-sqr-sqrt 99.0%

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

      6. hypot-def 99.2%

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

      7. pow1/2 99.2%

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

      8. sqrt-pow1 99.2%

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

      9. metadata-eval 99.2%

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

      10. pow1/2 99.2%

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

      11. sqrt-pow1 99.2%

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

      12. metadata-eval 99.2%

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

   8. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1}} \]
   9. Step-by-step derivation

      1. pow-sqr 99.0%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\left(2 \cdot -1\right)}} \]

      2. metadata-eval 99.0%

         \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   11. Taylor expanded in x around inf 98.1%

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

       1. unpow2 98.1%

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{1}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \]

       2. rem-square-sqrt 99.8%

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{1}{\color{blue}{2}} \]

       3. metadata-eval 99.8%

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

       4. *-commutative 99.8%

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

   13. Simplified 99.8%

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

       1. expm1-log1p-u 99.8%

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

       2. expm1-udef 4.7%

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

       3. inv-pow 4.7%

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

       4. sqrt-pow1 4.7%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

       5. metadata-eval 4.7%

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

   15. Applied egg-rr 4.7%

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

       1. expm1-def 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 100.0%

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

   17. Simplified 100.0%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. flip-- 58.1%

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

   2. div-inv 58.1%

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

   3. add-sqr-sqrt 58.1%

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

   4. add-sqr-sqrt 58.2%

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

   5. associate--l+ 58.3%

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

4. Applied egg-rr 58.3%

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

   1. +-commutative 58.3%

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

   2. associate-+l- 99.7%

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

   3. +-inverses 99.7%

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

   4. metadata-eval 99.7%

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

   5. associate-*r/ 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

6. Simplified 99.7%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 99.7%

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

   3. inv-pow 99.7%

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

   4. inv-pow 99.7%

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

   5. pow-prod-up 99.7%

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

   6. metadata-eval 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

   \[\leadsto \sqrt{{\left(\sqrt{1 + x} + \sqrt{x}\right)}^{-2}} \]
10. Add Preprocessing

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. flip-- 58.1%

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

   2. div-inv 58.1%

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

   3. add-sqr-sqrt 58.1%

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

   4. add-sqr-sqrt 58.2%

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

   5. associate--l+ 58.3%

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

4. Applied egg-rr 58.3%

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

   1. +-commutative 58.3%

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

   2. associate-+l- 99.7%

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

   3. +-inverses 99.7%

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

   4. metadata-eval 99.7%

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

   5. associate-*r/ 99.7%

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

   6. metadata-eval 99.7%

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

   7. +-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 98.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;1 + \left(\left(x \cdot \left(x \cdot -0.125\right) + x \cdot 0.5\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.22)
   (+ 1.0 (- (+ (* x (* x -0.125)) (* x 0.5)) (sqrt x)))
   (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.22) {
		tmp = 1.0 + (((x * (x * -0.125)) + (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 <= 1.22d0) then
        tmp = 1.0d0 + (((x * (x * (-0.125d0))) + (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 <= 1.22) {
		tmp = 1.0 + (((x * (x * -0.125)) + (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 <= 1.22:
		tmp = 1.0 + (((x * (x * -0.125)) + (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 <= 1.22)
		tmp = Float64(1.0 + Float64(Float64(Float64(x * Float64(x * -0.125)) + 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 <= 1.22)
		tmp = 1.0 + (((x * (x * -0.125)) + (x * 0.5)) - sqrt(x));
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.22], N[(1.0 + N[(N[(N[(x * N[(x * -0.125), $MachinePrecision]), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.21999999999999997

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. unpow2 99.2%

         \[\leadsto \left(1 + \left(0.5 \cdot x + -0.125 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) - \sqrt{x} \]

      3. associate-*r* 99.2%

         \[\leadsto \left(1 + \left(0.5 \cdot x + \color{blue}{\left(-0.125 \cdot x\right) \cdot x}\right)\right) - \sqrt{x} \]

      4. distribute-rgt-out 99.2%

         \[\leadsto \left(1 + \color{blue}{x \cdot \left(0.5 + -0.125 \cdot x\right)}\right) - \sqrt{x} \]

      5. *-commutative 99.2%

         \[\leadsto \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot -0.125}\right)\right) - \sqrt{x} \]

   5. Simplified 99.2%

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

      1. associate--l+ 99.2%

         \[\leadsto \color{blue}{1 + \left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right)} \]

      2. +-commutative 99.2%

         \[\leadsto \color{blue}{\left(x \cdot \left(0.5 + x \cdot -0.125\right) - \sqrt{x}\right) + 1} \]

      3. +-commutative 99.2%

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot -0.125 + 0.5\right)} - \sqrt{x}\right) + 1 \]

      4. fma-def 99.2%

         \[\leadsto \left(x \cdot \color{blue}{\mathsf{fma}\left(x, -0.125, 0.5\right)} - \sqrt{x}\right) + 1 \]

   7. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\left(x \cdot \mathsf{fma}\left(x, -0.125, 0.5\right) - \sqrt{x}\right) + 1} \]
   8. Step-by-step derivation

      1. fma-udef 99.2%

         \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot -0.125 + 0.5\right)} - \sqrt{x}\right) + 1 \]

      2. distribute-rgt-in 99.2%

         \[\leadsto \left(\color{blue}{\left(\left(x \cdot -0.125\right) \cdot x + 0.5 \cdot x\right)} - \sqrt{x}\right) + 1 \]

   9. Applied egg-rr 99.2%

      \[\leadsto \left(\color{blue}{\left(\left(x \cdot -0.125\right) \cdot x + 0.5 \cdot x\right)} - \sqrt{x}\right) + 1 \]

   if 1.21999999999999997 < x

   1. Initial program 4.5%

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

      1. flip-- 4.5%

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

      2. div-inv 4.5%

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

      3. add-sqr-sqrt 4.5%

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

      4. add-sqr-sqrt 4.7%

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

      5. associate--l+ 4.7%

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

   4. Applied egg-rr 4.7%

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

      1. +-commutative 4.7%

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

      2. associate-+l- 99.5%

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

      3. +-inverses 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate-*r/ 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. inv-pow 99.5%

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

      2. add-sqr-sqrt 99.2%

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

      3. unpow-prod-down 99.1%

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

      4. add-sqr-sqrt 99.0%

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

      5. add-sqr-sqrt 99.0%

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

      6. hypot-def 99.1%

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

      7. pow1/2 99.1%

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

      8. sqrt-pow1 99.2%

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

      9. metadata-eval 99.2%

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

      10. pow1/2 99.2%

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

      11. sqrt-pow1 99.2%

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

      12. metadata-eval 99.2%

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

   8. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1}} \]
   9. Step-by-step derivation

      1. pow-sqr 99.0%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\left(2 \cdot -1\right)}} \]

      2. metadata-eval 99.0%

         \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   11. Taylor expanded in x around inf 97.6%

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

       1. unpow2 97.6%

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{1}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \]

       2. rem-square-sqrt 99.4%

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{1}{\color{blue}{2}} \]

       3. metadata-eval 99.4%

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

       4. *-commutative 99.4%

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

   13. Simplified 99.4%

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

       1. expm1-log1p-u 99.4%

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

       2. expm1-udef 5.1%

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

       3. inv-pow 5.1%

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

       4. sqrt-pow1 5.1%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

       5. metadata-eval 5.1%

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

   15. Applied egg-rr 5.1%

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

       1. expm1-def 99.5%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 99.5%

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

   17. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;1 + \left(\left(x \cdot \left(x \cdot -0.125\right) + x \cdot 0.5\right) - \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.22)
   (- (+ 1.0 (* x (+ 0.5 (* x -0.125)))) (sqrt x))
   (* 0.5 (pow x -0.5))))

C

double code(double x) {
	double tmp;
	if (x <= 1.22) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - 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 <= 1.22d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * (-0.125d0))))) - 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 <= 1.22) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - Math.sqrt(x);
	} else {
		tmp = 0.5 * Math.pow(x, -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.22:
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - math.sqrt(x)
	else:
		tmp = 0.5 * math.pow(x, -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.22)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * -0.125)))) - 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 <= 1.22)
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - sqrt(x);
	else
		tmp = 0.5 * (x ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.22], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $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 < 1.21999999999999997

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. unpow2 99.2%

         \[\leadsto \left(1 + \left(0.5 \cdot x + -0.125 \cdot \color{blue}{\left(x \cdot x\right)}\right)\right) - \sqrt{x} \]

      3. associate-*r* 99.2%

         \[\leadsto \left(1 + \left(0.5 \cdot x + \color{blue}{\left(-0.125 \cdot x\right) \cdot x}\right)\right) - \sqrt{x} \]

      4. distribute-rgt-out 99.2%

         \[\leadsto \left(1 + \color{blue}{x \cdot \left(0.5 + -0.125 \cdot x\right)}\right) - \sqrt{x} \]

      5. *-commutative 99.2%

         \[\leadsto \left(1 + x \cdot \left(0.5 + \color{blue}{x \cdot -0.125}\right)\right) - \sqrt{x} \]

   5. Simplified 99.2%

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right)} - \sqrt{x} \]

   if 1.21999999999999997 < x

   1. Initial program 4.5%

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

      1. flip-- 4.5%

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

      2. div-inv 4.5%

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

      3. add-sqr-sqrt 4.5%

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

      4. add-sqr-sqrt 4.7%

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

      5. associate--l+ 4.7%

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

   4. Applied egg-rr 4.7%

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

      1. +-commutative 4.7%

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

      2. associate-+l- 99.5%

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

      3. +-inverses 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate-*r/ 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. inv-pow 99.5%

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

      2. add-sqr-sqrt 99.2%

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

      3. unpow-prod-down 99.1%

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

      4. add-sqr-sqrt 99.0%

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

      5. add-sqr-sqrt 99.0%

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

      6. hypot-def 99.1%

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

      7. pow1/2 99.1%

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

      8. sqrt-pow1 99.2%

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

      9. metadata-eval 99.2%

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

      10. pow1/2 99.2%

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

      11. sqrt-pow1 99.2%

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

      12. metadata-eval 99.2%

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

   8. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1}} \]
   9. Step-by-step derivation

      1. pow-sqr 99.0%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\left(2 \cdot -1\right)}} \]

      2. metadata-eval 99.0%

         \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   11. Taylor expanded in x around inf 97.6%

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

       1. unpow2 97.6%

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{1}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \]

       2. rem-square-sqrt 99.4%

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{1}{\color{blue}{2}} \]

       3. metadata-eval 99.4%

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

       4. *-commutative 99.4%

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

   13. Simplified 99.4%

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

       1. expm1-log1p-u 99.4%

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

       2. expm1-udef 5.1%

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

       3. inv-pow 5.1%

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

       4. sqrt-pow1 5.1%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

       5. metadata-eval 5.1%

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

   15. Applied egg-rr 5.1%

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

       1. expm1-def 99.5%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 99.5%

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

   17. Simplified 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.22:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {x}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 99.9%

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

   3. Taylor expanded in x around 0 99.0%

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

   if 1 < x

   1. Initial program 4.5%

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

      1. flip-- 4.5%

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

      2. div-inv 4.5%

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

      3. add-sqr-sqrt 4.5%

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

      4. add-sqr-sqrt 4.7%

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

      5. associate--l+ 4.7%

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

   4. Applied egg-rr 4.7%

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

      1. +-commutative 4.7%

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

      2. associate-+l- 99.5%

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

      3. +-inverses 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate-*r/ 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. inv-pow 99.5%

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

      2. add-sqr-sqrt 99.2%

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

      3. unpow-prod-down 99.1%

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

      4. add-sqr-sqrt 99.0%

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

      5. add-sqr-sqrt 99.0%

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

      6. hypot-def 99.1%

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

      7. pow1/2 99.1%

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

      8. sqrt-pow1 99.2%

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

      9. metadata-eval 99.2%

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

      10. pow1/2 99.2%

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

      11. sqrt-pow1 99.2%

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

      12. metadata-eval 99.2%

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

   8. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1}} \]
   9. Step-by-step derivation

      1. pow-sqr 99.0%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\left(2 \cdot -1\right)}} \]

      2. metadata-eval 99.0%

         \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   11. Taylor expanded in x around inf 97.6%

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

       1. unpow2 97.6%

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{1}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \]

       2. rem-square-sqrt 99.4%

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{1}{\color{blue}{2}} \]

       3. metadata-eval 99.4%

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

       4. *-commutative 99.4%

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

   13. Simplified 99.4%

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

       1. expm1-log1p-u 99.4%

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

       2. expm1-udef 5.1%

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

       3. inv-pow 5.1%

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

       4. sqrt-pow1 5.1%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

       5. metadata-eval 5.1%

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

   15. Applied egg-rr 5.1%

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

       1. expm1-def 99.5%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 99.5%

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

   17. Simplified 99.5%

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

4. Final simplification 99.2%

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

Alternative 7: 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 99.9%

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

      1. flip-- 99.8%

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

      2. div-inv 99.8%

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

      3. add-sqr-sqrt 99.9%

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

      4. add-sqr-sqrt 99.9%

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

      5. associate--l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l- 99.9%

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

      3. +-inverses 99.9%

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

      4. metadata-eval 99.9%

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

      5. associate-*r/ 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   6. Simplified 99.9%

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

      1. inv-pow 99.9%

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

      2. add-sqr-sqrt 99.8%

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

      3. unpow-prod-down 99.8%

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

      4. add-sqr-sqrt 99.8%

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

      5. add-sqr-sqrt 99.8%

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

      6. hypot-def 99.8%

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

      7. pow1/2 99.8%

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

      8. sqrt-pow1 99.8%

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

      9. metadata-eval 99.8%

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

      10. pow1/2 99.8%

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

      11. sqrt-pow1 99.8%

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

      12. metadata-eval 99.8%

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

   8. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1}} \]
   9. Step-by-step derivation

      1. pow-sqr 99.8%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\left(2 \cdot -1\right)}} \]

      2. metadata-eval 99.8%

         \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   10. Simplified 99.8%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   11. Taylor expanded in x around 0 97.9%

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

   if 1 < x

   1. Initial program 4.5%

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

      1. flip-- 4.5%

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

      2. div-inv 4.5%

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

      3. add-sqr-sqrt 4.5%

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

      4. add-sqr-sqrt 4.7%

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

      5. associate--l+ 4.7%

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

   4. Applied egg-rr 4.7%

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

      1. +-commutative 4.7%

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

      2. associate-+l- 99.5%

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

      3. +-inverses 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate-*r/ 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. inv-pow 99.5%

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

      2. add-sqr-sqrt 99.2%

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

      3. unpow-prod-down 99.1%

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

      4. add-sqr-sqrt 99.0%

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

      5. add-sqr-sqrt 99.0%

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

      6. hypot-def 99.1%

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

      7. pow1/2 99.1%

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

      8. sqrt-pow1 99.2%

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

      9. metadata-eval 99.2%

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

      10. pow1/2 99.2%

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

      11. sqrt-pow1 99.2%

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

      12. metadata-eval 99.2%

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

   8. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1}} \]
   9. Step-by-step derivation

      1. pow-sqr 99.0%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\left(2 \cdot -1\right)}} \]

      2. metadata-eval 99.0%

         \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   11. Taylor expanded in x around inf 97.6%

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

       1. unpow2 97.6%

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{1}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \]

       2. rem-square-sqrt 99.4%

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{1}{\color{blue}{2}} \]

       3. metadata-eval 99.4%

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

       4. *-commutative 99.4%

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

   13. Simplified 99.4%

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

       1. expm1-log1p-u 99.4%

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

       2. expm1-udef 5.1%

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

       3. inv-pow 5.1%

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

       4. sqrt-pow1 5.1%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

       5. metadata-eval 5.1%

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

   15. Applied egg-rr 5.1%

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

       1. expm1-def 99.5%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 99.5%

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

   17. Simplified 99.5%

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

4. Final simplification 98.6%

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

Alternative 8: 96.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.25

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 96.3%

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

   if 0.25 < x

   1. Initial program 5.3%

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

      1. flip-- 5.3%

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

      2. div-inv 5.3%

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

      3. add-sqr-sqrt 5.3%

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

      4. add-sqr-sqrt 5.6%

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

      5. associate--l+ 5.6%

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

   4. Applied egg-rr 5.6%

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

      1. +-commutative 5.6%

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

      2. associate-+l- 99.5%

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

      3. +-inverses 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate-*r/ 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. inv-pow 99.5%

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

      2. add-sqr-sqrt 99.2%

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

      3. unpow-prod-down 99.1%

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

      4. add-sqr-sqrt 99.0%

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

      5. add-sqr-sqrt 99.0%

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

      6. hypot-def 99.1%

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

      7. pow1/2 99.1%

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

      8. sqrt-pow1 99.2%

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

      9. metadata-eval 99.2%

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

      10. pow1/2 99.2%

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

      11. sqrt-pow1 99.2%

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

      12. metadata-eval 99.2%

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

   8. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1}} \]
   9. Step-by-step derivation

      1. pow-sqr 99.0%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\left(2 \cdot -1\right)}} \]

      2. metadata-eval 99.0%

         \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   11. Taylor expanded in x around inf 96.9%

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

       1. unpow2 96.9%

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{1}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}} \]

       2. rem-square-sqrt 98.7%

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \frac{1}{\color{blue}{2}} \]

       3. metadata-eval 98.7%

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

       4. *-commutative 98.7%

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

   13. Simplified 98.7%

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

       1. expm1-log1p-u 98.7%

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

       2. expm1-udef 5.2%

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

       3. inv-pow 5.2%

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

       4. sqrt-pow1 5.2%

          \[\leadsto 0.5 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

       5. metadata-eval 5.2%

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

   15. Applied egg-rr 5.2%

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

       1. expm1-def 98.8%

          \[\leadsto 0.5 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 98.8%

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

   17. Simplified 98.8%

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

4. Final simplification 97.4%

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

Alternative 9: 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 100.0%

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

   3. Taylor expanded in x around 0 96.3%

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

   if 0.25 < x

   1. Initial program 5.3%

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

      1. flip-- 5.3%

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

      2. div-inv 5.3%

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

      3. add-sqr-sqrt 5.3%

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

      4. add-sqr-sqrt 5.6%

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

      5. associate--l+ 5.6%

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

   4. Applied egg-rr 5.6%

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

      1. +-commutative 5.6%

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

      2. associate-+l- 99.5%

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

      3. +-inverses 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate-*r/ 99.5%

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

      6. metadata-eval 99.5%

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

      7. +-commutative 99.5%

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

   6. Simplified 99.5%

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

      1. inv-pow 99.5%

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

      2. add-sqr-sqrt 99.2%

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

      3. unpow-prod-down 99.1%

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

      4. add-sqr-sqrt 99.0%

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

      5. add-sqr-sqrt 99.0%

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

      6. hypot-def 99.1%

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

      7. pow1/2 99.1%

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

      8. sqrt-pow1 99.2%

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

      9. metadata-eval 99.2%

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

      10. pow1/2 99.2%

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

      11. sqrt-pow1 99.2%

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

      12. metadata-eval 99.2%

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

   8. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1} \cdot {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-1}} \]
   9. Step-by-step derivation

      1. pow-sqr 99.0%

         \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\left(2 \cdot -1\right)}} \]

      2. metadata-eval 99.0%

         \[\leadsto {\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{\color{blue}{-2}} \]

   10. Simplified 99.0%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left({\left(1 + x\right)}^{0.25}, {x}^{0.25}\right)\right)}^{-2}} \]

   11. Taylor expanded in x around inf 97.8%

       \[\leadsto {\left(\mathsf{hypot}\left(\color{blue}{{x}^{0.25}}, {x}^{0.25}\right)\right)}^{-2} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 97.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left({x}^{0.25}, {x}^{0.25}\right)\right)}^{-2}\right)\right)} \]

       2. expm1-udef 5.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left({x}^{0.25}, {x}^{0.25}\right)\right)}^{-2}\right)} - 1} \]

       3. hypot-udef 5.2%

          \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(\sqrt{{x}^{0.25} \cdot {x}^{0.25} + {x}^{0.25} \cdot {x}^{0.25}}\right)}}^{-2}\right)} - 1 \]

       4. sqrt-pow2 5.2%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left({x}^{0.25} \cdot {x}^{0.25} + {x}^{0.25} \cdot {x}^{0.25}\right)}^{\left(\frac{-2}{2}\right)}}\right)} - 1 \]

       5. pow-prod-up 5.2%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\color{blue}{{x}^{\left(0.25 + 0.25\right)}} + {x}^{0.25} \cdot {x}^{0.25}\right)}^{\left(\frac{-2}{2}\right)}\right)} - 1 \]

       6. metadata-eval 5.2%

          \[\leadsto e^{\mathsf{log1p}\left({\left({x}^{\color{blue}{0.5}} + {x}^{0.25} \cdot {x}^{0.25}\right)}^{\left(\frac{-2}{2}\right)}\right)} - 1 \]

       7. pow1/2 5.2%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\color{blue}{\sqrt{x}} + {x}^{0.25} \cdot {x}^{0.25}\right)}^{\left(\frac{-2}{2}\right)}\right)} - 1 \]

       8. pow-prod-up 5.2%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\sqrt{x} + \color{blue}{{x}^{\left(0.25 + 0.25\right)}}\right)}^{\left(\frac{-2}{2}\right)}\right)} - 1 \]

       9. metadata-eval 5.2%

          \[\leadsto e^{\mathsf{log1p}\left({\left(\sqrt{x} + {x}^{\color{blue}{0.5}}\right)}^{\left(\frac{-2}{2}\right)}\right)} - 1 \]

       10. pow1/2 5.2%

           \[\leadsto e^{\mathsf{log1p}\left({\left(\sqrt{x} + \color{blue}{\sqrt{x}}\right)}^{\left(\frac{-2}{2}\right)}\right)} - 1 \]

       11. count-2 5.2%

           \[\leadsto e^{\mathsf{log1p}\left({\color{blue}{\left(2 \cdot \sqrt{x}\right)}}^{\left(\frac{-2}{2}\right)}\right)} - 1 \]

       12. metadata-eval 5.2%

           \[\leadsto e^{\mathsf{log1p}\left({\left(2 \cdot \sqrt{x}\right)}^{\color{blue}{-1}}\right)} - 1 \]

   13. Applied egg-rr 5.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(2 \cdot \sqrt{x}\right)}^{-1}\right)} - 1} \]
   14. Step-by-step derivation

       1. expm1-def 98.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(2 \cdot \sqrt{x}\right)}^{-1}\right)\right)} \]

       2. expm1-log1p 98.3%

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

       3. unpow-1 98.3%

          \[\leadsto \color{blue}{\frac{1}{2 \cdot \sqrt{x}}} \]

       4. associate-/r* 98.3%

          \[\leadsto \color{blue}{\frac{\frac{1}{2}}{\sqrt{x}}} \]

       5. metadata-eval 98.3%

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

   15. Simplified 98.3%

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

4. Final simplification 97.2%

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

Alternative 10: 52.4% 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 58.2%

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

3. Taylor expanded in x around 0 56.7%

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

4. Final simplification 56.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024010 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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