Main:bigenough3 from C

Percentage Accurate: 54.0% → 99.8%

Time: 8.5s

Alternatives: 12

Speedup: 1.8×

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: 54.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

   1. sub-neg 56.8%

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

   2. flip-+ 57.2%

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

   3. add-sqr-sqrt 57.1%

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

   4. pow1 57.1%

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

   5. pow1 57.1%

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

   6. pow-prod-up 57.1%

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

   7. metadata-eval 57.1%

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

4. Applied egg-rr 57.1%

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

   1. unpow2 57.1%

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

   2. sqr-neg 57.1%

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

   3. add-sqr-sqrt 57.4%

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

   4. *-un-lft-identity 57.4%

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

   5. +-commutative 57.4%

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

   6. *-un-lft-identity 57.4%

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

   7. associate--l+ 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \frac{1 + \left(x - x\right)}{\sqrt{1 + x} + \sqrt{x}} \]
8. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;t\_0 - \sqrt{x} \leq 0:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{t\_0 + \sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (- t_0 (sqrt x)) 0.0)
     (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt x)))
     (/ (+ x (- 1.0 x)) (+ t_0 (sqrt x))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if ((t_0 - sqrt(x)) <= 0.0) {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt(x));
	} else {
		tmp = (x + (1.0 - x)) / (t_0 + sqrt(x));
	}
	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))
    if ((t_0 - sqrt(x)) <= 0.0d0) then
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + sqrt(x))
    else
        tmp = (x + (1.0d0 - x)) / (t_0 + sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if ((t_0 - Math.sqrt(x)) <= 0.0) {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt(x));
	} else {
		tmp = (x + (1.0 - x)) / (t_0 + Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if (t_0 - math.sqrt(x)) <= 0.0:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt(x))
	else:
		tmp = (x + (1.0 - x)) / (t_0 + math.sqrt(x))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - sqrt(x)) <= 0.0)
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(x)));
	else
		tmp = Float64(Float64(x + Float64(1.0 - x)) / Float64(t_0 + sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if ((t_0 - sqrt(x)) <= 0.0)
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt(x));
	else
		tmp = (x + (1.0 - x)) / (t_0 + sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.8%

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

      1. sub-neg 3.8%

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

      2. flip-+ 3.8%

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

      3. add-sqr-sqrt 3.5%

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

      4. pow1 3.5%

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

      5. pow1 3.5%

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

      6. pow-prod-up 3.5%

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

      7. metadata-eval 3.5%

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

   4. Applied egg-rr 3.5%

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

      1. unpow2 3.5%

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

      2. sqr-neg 3.5%

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

      3. add-sqr-sqrt 3.8%

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

      4. *-un-lft-identity 3.8%

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

      5. +-commutative 3.8%

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

      6. *-un-lft-identity 3.8%

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

      7. associate--l+ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

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

      4. sqrt-pow1 99.3%

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

      5. metadata-eval 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in x around inf 99.6%

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

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

   1. Initial program 98.8%

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

      1. flip-- 99.4%

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

      2. add-sqr-sqrt 99.3%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate--l+ 99.9%

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

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{x + \left(1 - x\right)}{\sqrt{x + 1} + \sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt{1 + x} + \sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;t\_0 - \sqrt{x} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{x}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (- t_0 (sqrt x)) 2e-7)
     (/ (+ 1.0 (- x x)) (+ (sqrt x) (sqrt x)))
     (- t_0 (/ x (sqrt x))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if ((t_0 - sqrt(x)) <= 2e-7) {
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt(x));
	} else {
		tmp = t_0 - (x / sqrt(x));
	}
	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))
    if ((t_0 - sqrt(x)) <= 2d-7) then
        tmp = (1.0d0 + (x - x)) / (sqrt(x) + sqrt(x))
    else
        tmp = t_0 - (x / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if ((t_0 - Math.sqrt(x)) <= 2e-7) {
		tmp = (1.0 + (x - x)) / (Math.sqrt(x) + Math.sqrt(x));
	} else {
		tmp = t_0 - (x / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if (t_0 - math.sqrt(x)) <= 2e-7:
		tmp = (1.0 + (x - x)) / (math.sqrt(x) + math.sqrt(x))
	else:
		tmp = t_0 - (x / math.sqrt(x))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - sqrt(x)) <= 2e-7)
		tmp = Float64(Float64(1.0 + Float64(x - x)) / Float64(sqrt(x) + sqrt(x)));
	else
		tmp = Float64(t_0 - Float64(x / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if ((t_0 - sqrt(x)) <= 2e-7)
		tmp = (1.0 + (x - x)) / (sqrt(x) + sqrt(x));
	else
		tmp = t_0 - (x / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 2e-7], N[(N[(1.0 + N[(x - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(x / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x)) < 1.9999999999999999e-7

   1. Initial program 4.2%

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

      1. sub-neg 4.2%

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

      2. flip-+ 4.9%

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

      3. add-sqr-sqrt 4.6%

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

      4. pow1 4.6%

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

      5. pow1 4.6%

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

      6. pow-prod-up 4.6%

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

      7. metadata-eval 4.6%

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

   4. Applied egg-rr 4.6%

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

      1. unpow2 4.6%

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

      2. sqr-neg 4.6%

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

      3. add-sqr-sqrt 5.4%

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

      4. *-un-lft-identity 5.4%

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

      5. +-commutative 5.4%

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

      6. *-un-lft-identity 5.4%

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

      7. associate--l+ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. add-sqr-sqrt 99.3%

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

      2. pow2 99.3%

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

      3. pow1/2 99.3%

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

      4. sqrt-pow1 99.3%

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

      5. metadata-eval 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in x around inf 99.4%

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

   if 1.9999999999999999e-7 < (-.f64 (sqrt.f64 (+.f64 x 1)) (sqrt.f64 x))

   1. Initial program 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. sub-neg 99.7%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(-\sqrt{x}\right)\right)} + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right) \]

      2. associate-+l+ 99.7%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)\right)} \]

      3. fma-undefine 99.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x} + \sqrt{x}\right)}\right) \]

      4. +-commutative 99.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\sqrt{x} + \left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x}\right)}\right) \]

      5. *-commutative 99.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\sqrt[3]{x} \cdot \left(-{x}^{0.16666666666666666}\right)}\right)\right) \]

      6. distribute-rgt-neg-out 99.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)}\right)\right) \]

      7. distribute-lft-neg-out 99.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}}\right)\right) \]

      8. associate-+r+ 99.7%

         \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\left(-\sqrt{x}\right) + \sqrt{x}\right) + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right)} \]

      9. +-commutative 99.7%

         \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} + \left(-\sqrt{x}\right)\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      10. unsub-neg 99.7%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} - \sqrt{x}\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      11. +-inverses 99.7%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{0} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      12. +-commutative 99.7%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666} + 0\right)} \]

      13. +-rgt-identity 99.7%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}} \]

      14. distribute-lft-neg-out 99.7%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)} \]

      15. unsub-neg 99.7%

          \[\leadsto \color{blue}{\sqrt{x + 1} - \sqrt[3]{x} \cdot {x}^{0.16666666666666666}} \]

      16. *-commutative 99.7%

          \[\leadsto \sqrt{x + 1} - \color{blue}{{x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} - {x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]
   7. Step-by-step derivation

      1. pow1/3 99.7%

         \[\leadsto \sqrt{1 + x} - {x}^{0.16666666666666666} \cdot \color{blue}{{x}^{0.3333333333333333}} \]

      2. pow-prod-up 99.7%

         \[\leadsto \sqrt{1 + x} - \color{blue}{{x}^{\left(0.16666666666666666 + 0.3333333333333333\right)}} \]

      3. metadata-eval 99.7%

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

      4. pow1/2 99.7%

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

      5. add-sqr-sqrt 99.7%

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

      6. sqr-neg 99.7%

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

      7. sqrt-prod 0.0%

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

      8. add-sqr-sqrt 91.8%

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

      9. neg-sub0 91.8%

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

      10. flip-- 91.8%

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

      11. metadata-eval 91.8%

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

      12. add-sqr-sqrt 91.8%

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

      13. neg-sub0 91.8%

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

      14. add-sqr-sqrt 91.8%

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

      15. distribute-lft-neg-in 91.8%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-prod 99.8%

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

      18. sqr-neg 99.8%

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

      19. add-sqr-sqrt 99.8%

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

      20. add-sqr-sqrt 99.8%

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

      21. add-sqr-sqrt 99.8%

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

      22. sqr-neg 99.8%

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

      23. sqrt-prod 0.0%

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

      24. add-sqr-sqrt 91.8%

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

      25. sub-neg 91.8%

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

      26. neg-sub0 91.8%

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

      27. add-sqr-sqrt 0.0%

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;\frac{1 + \left(x - x\right)}{\sqrt{x} + \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \frac{x}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;t\_0 - \sqrt{x} \leq 0:\\ \;\;\;\;\frac{-1}{-\sqrt{1 + \left(x + x\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{x}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (- t_0 (sqrt x)) 0.0)
     (/ -1.0 (- (sqrt (+ 1.0 (+ x x)))))
     (- t_0 (/ x (sqrt x))))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if ((t_0 - sqrt(x)) <= 0.0) {
		tmp = -1.0 / -sqrt((1.0 + (x + x)));
	} else {
		tmp = t_0 - (x / sqrt(x));
	}
	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))
    if ((t_0 - sqrt(x)) <= 0.0d0) then
        tmp = (-1.0d0) / -sqrt((1.0d0 + (x + x)))
    else
        tmp = t_0 - (x / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if ((t_0 - Math.sqrt(x)) <= 0.0) {
		tmp = -1.0 / -Math.sqrt((1.0 + (x + x)));
	} else {
		tmp = t_0 - (x / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if (t_0 - math.sqrt(x)) <= 0.0:
		tmp = -1.0 / -math.sqrt((1.0 + (x + x)))
	else:
		tmp = t_0 - (x / math.sqrt(x))
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(t_0 - sqrt(x)) <= 0.0)
		tmp = Float64(-1.0 / Float64(-sqrt(Float64(1.0 + Float64(x + x)))));
	else
		tmp = Float64(t_0 - Float64(x / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if ((t_0 - sqrt(x)) <= 0.0)
		tmp = -1.0 / -sqrt((1.0 + (x + x)));
	else
		tmp = t_0 - (x / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.0], N[(-1.0 / (-N[Sqrt[N[(1.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(t$95$0 - N[(x / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 3.8%

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

   4. Applied egg-rr 5.8%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. sub-neg 5.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(-\sqrt{x}\right)\right)} + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right) \]

      2. associate-+l+ 5.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)\right)} \]

      3. fma-undefine 5.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x} + \sqrt{x}\right)}\right) \]

      4. +-commutative 5.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\sqrt{x} + \left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x}\right)}\right) \]

      5. *-commutative 5.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\sqrt[3]{x} \cdot \left(-{x}^{0.16666666666666666}\right)}\right)\right) \]

      6. distribute-rgt-neg-out 5.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)}\right)\right) \]

      7. distribute-lft-neg-out 5.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}}\right)\right) \]

      8. associate-+r+ 5.8%

         \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\left(-\sqrt{x}\right) + \sqrt{x}\right) + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right)} \]

      9. +-commutative 5.8%

         \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} + \left(-\sqrt{x}\right)\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      10. unsub-neg 5.8%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} - \sqrt{x}\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      11. +-inverses 5.8%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{0} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      12. +-commutative 5.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666} + 0\right)} \]

      13. +-rgt-identity 5.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}} \]

      14. distribute-lft-neg-out 5.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)} \]

      15. unsub-neg 5.8%

          \[\leadsto \color{blue}{\sqrt{x + 1} - \sqrt[3]{x} \cdot {x}^{0.16666666666666666}} \]

      16. *-commutative 5.8%

          \[\leadsto \sqrt{x + 1} - \color{blue}{{x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

   6. Simplified 5.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} - {x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

   8. Applied egg-rr 3.8%

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

   9. Taylor expanded in x around 0 20.3%

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

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

   1. Initial program 98.8%

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

   4. Applied egg-rr 98.7%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. sub-neg 98.7%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(-\sqrt{x}\right)\right)} + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right) \]

      2. associate-+l+ 98.7%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)\right)} \]

      3. fma-undefine 98.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x} + \sqrt{x}\right)}\right) \]

      4. +-commutative 98.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\sqrt{x} + \left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x}\right)}\right) \]

      5. *-commutative 98.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\sqrt[3]{x} \cdot \left(-{x}^{0.16666666666666666}\right)}\right)\right) \]

      6. distribute-rgt-neg-out 98.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)}\right)\right) \]

      7. distribute-lft-neg-out 98.7%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}}\right)\right) \]

      8. associate-+r+ 98.7%

         \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\left(-\sqrt{x}\right) + \sqrt{x}\right) + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right)} \]

      9. +-commutative 98.7%

         \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} + \left(-\sqrt{x}\right)\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      10. unsub-neg 98.7%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} - \sqrt{x}\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      11. +-inverses 98.7%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{0} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      12. +-commutative 98.7%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666} + 0\right)} \]

      13. +-rgt-identity 98.7%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}} \]

      14. distribute-lft-neg-out 98.7%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)} \]

      15. unsub-neg 98.7%

          \[\leadsto \color{blue}{\sqrt{x + 1} - \sqrt[3]{x} \cdot {x}^{0.16666666666666666}} \]

      16. *-commutative 98.7%

          \[\leadsto \sqrt{x + 1} - \color{blue}{{x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

   6. Simplified 98.7%

      \[\leadsto \color{blue}{\sqrt{1 + x} - {x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]
   7. Step-by-step derivation

      1. pow1/3 98.7%

         \[\leadsto \sqrt{1 + x} - {x}^{0.16666666666666666} \cdot \color{blue}{{x}^{0.3333333333333333}} \]

      2. pow-prod-up 98.8%

         \[\leadsto \sqrt{1 + x} - \color{blue}{{x}^{\left(0.16666666666666666 + 0.3333333333333333\right)}} \]

      3. metadata-eval 98.8%

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

      4. pow1/2 98.8%

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

      5. add-sqr-sqrt 98.8%

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

      6. sqr-neg 98.8%

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

      7. sqrt-prod 0.0%

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

      8. add-sqr-sqrt 90.6%

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

      9. neg-sub0 90.6%

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

      10. flip-- 90.6%

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

      11. metadata-eval 90.6%

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

      12. add-sqr-sqrt 90.6%

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

      13. neg-sub0 90.6%

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

      14. add-sqr-sqrt 90.6%

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

      15. distribute-lft-neg-in 90.6%

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

      16. add-sqr-sqrt 0.0%

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

      17. sqrt-prod 98.8%

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

      18. sqr-neg 98.8%

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

      19. add-sqr-sqrt 98.8%

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

      20. add-sqr-sqrt 98.8%

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

      21. add-sqr-sqrt 98.8%

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

      22. sqr-neg 98.8%

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

      23. sqrt-prod 0.0%

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

      24. add-sqr-sqrt 90.6%

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

      25. sub-neg 90.6%

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

      26. neg-sub0 90.6%

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

      27. add-sqr-sqrt 0.0%

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

   8. Applied egg-rr 98.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0:\\ \;\;\;\;\frac{-1}{-\sqrt{1 + \left(x + x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \frac{x}{\sqrt{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-1}{-\sqrt{1 + \left(x + x\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ 1.0 x)) (sqrt x))))
   (if (<= t_0 0.0) (/ -1.0 (- (sqrt (+ 1.0 (+ x x))))) t_0)))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -1.0 / -sqrt((1.0 + (x + x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 0.0d0) then
        tmp = (-1.0d0) / -sqrt((1.0d0 + (x + x)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -1.0 / -Math.sqrt((1.0 + (x + x)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 0.0:
		tmp = -1.0 / -math.sqrt((1.0 + (x + x)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(-1.0 / Float64(-sqrt(Float64(1.0 + Float64(x + x)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = -1.0 / -sqrt((1.0 + (x + x)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(-1.0 / (-N[Sqrt[N[(1.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]], $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 3.8%

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

   4. Applied egg-rr 5.8%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. sub-neg 5.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(-\sqrt{x}\right)\right)} + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right) \]

      2. associate-+l+ 5.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)\right)} \]

      3. fma-undefine 5.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x} + \sqrt{x}\right)}\right) \]

      4. +-commutative 5.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\sqrt{x} + \left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x}\right)}\right) \]

      5. *-commutative 5.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\sqrt[3]{x} \cdot \left(-{x}^{0.16666666666666666}\right)}\right)\right) \]

      6. distribute-rgt-neg-out 5.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)}\right)\right) \]

      7. distribute-lft-neg-out 5.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}}\right)\right) \]

      8. associate-+r+ 5.8%

         \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\left(-\sqrt{x}\right) + \sqrt{x}\right) + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right)} \]

      9. +-commutative 5.8%

         \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} + \left(-\sqrt{x}\right)\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      10. unsub-neg 5.8%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} - \sqrt{x}\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      11. +-inverses 5.8%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{0} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      12. +-commutative 5.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666} + 0\right)} \]

      13. +-rgt-identity 5.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}} \]

      14. distribute-lft-neg-out 5.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)} \]

      15. unsub-neg 5.8%

          \[\leadsto \color{blue}{\sqrt{x + 1} - \sqrt[3]{x} \cdot {x}^{0.16666666666666666}} \]

      16. *-commutative 5.8%

          \[\leadsto \sqrt{x + 1} - \color{blue}{{x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

   6. Simplified 5.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} - {x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

   8. Applied egg-rr 3.8%

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

   9. Taylor expanded in x around 0 20.3%

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

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

   1. Initial program 98.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 + x} - \sqrt{x} \leq 0:\\ \;\;\;\;\frac{-1}{-\sqrt{1 + \left(x + x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + x} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt{x} + \left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sqrt{1 + \left(x + x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.0)
   (/ (+ x (- 1.0 x)) (+ (sqrt x) (+ 1.0 (* x (+ 0.5 (* x -0.125))))))
   (/ -1.0 (- (sqrt (+ 1.0 (+ x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (x + (1.0 - x)) / (sqrt(x) + (1.0 + (x * (0.5 + (x * -0.125)))));
	} else {
		tmp = -1.0 / -sqrt((1.0 + (x + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.0d0) then
        tmp = (x + (1.0d0 - x)) / (sqrt(x) + (1.0d0 + (x * (0.5d0 + (x * (-0.125d0))))))
    else
        tmp = (-1.0d0) / -sqrt((1.0d0 + (x + x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = (x + (1.0 - x)) / (Math.sqrt(x) + (1.0 + (x * (0.5 + (x * -0.125)))));
	} else {
		tmp = -1.0 / -Math.sqrt((1.0 + (x + x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.0:
		tmp = (x + (1.0 - x)) / (math.sqrt(x) + (1.0 + (x * (0.5 + (x * -0.125)))))
	else:
		tmp = -1.0 / -math.sqrt((1.0 + (x + x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(Float64(x + Float64(1.0 - x)) / Float64(sqrt(x) + Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * -0.125))))));
	else
		tmp = Float64(-1.0 / Float64(-sqrt(Float64(1.0 + Float64(x + x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.0)
		tmp = (x + (1.0 - x)) / (sqrt(x) + (1.0 + (x * (0.5 + (x * -0.125)))));
	else
		tmp = -1.0 / -sqrt((1.0 + (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.0], N[(N[(x + N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + N[(1.0 + N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / (-N[Sqrt[N[(1.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   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. add-sqr-sqrt 99.9%

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

      3. add-sqr-sqrt 99.9%

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

      4. associate--l+ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 97.4%

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

      1. +-commutative 97.4%

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

      2. unpow2 97.4%

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

      3. associate-*r* 97.4%

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

      4. distribute-rgt-out 97.4%

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

      5. *-commutative 97.4%

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

   7. Simplified 97.4%

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

   if 4 < x

   1. Initial program 4.9%

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

   4. Applied egg-rr 6.8%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. sub-neg 6.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(-\sqrt{x}\right)\right)} + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right) \]

      2. associate-+l+ 6.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)\right)} \]

      3. fma-undefine 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x} + \sqrt{x}\right)}\right) \]

      4. +-commutative 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\sqrt{x} + \left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x}\right)}\right) \]

      5. *-commutative 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\sqrt[3]{x} \cdot \left(-{x}^{0.16666666666666666}\right)}\right)\right) \]

      6. distribute-rgt-neg-out 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)}\right)\right) \]

      7. distribute-lft-neg-out 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}}\right)\right) \]

      8. associate-+r+ 6.8%

         \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\left(-\sqrt{x}\right) + \sqrt{x}\right) + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right)} \]

      9. +-commutative 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} + \left(-\sqrt{x}\right)\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      10. unsub-neg 6.8%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} - \sqrt{x}\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      11. +-inverses 6.8%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{0} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      12. +-commutative 6.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666} + 0\right)} \]

      13. +-rgt-identity 6.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}} \]

      14. distribute-lft-neg-out 6.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)} \]

      15. unsub-neg 6.8%

          \[\leadsto \color{blue}{\sqrt{x + 1} - \sqrt[3]{x} \cdot {x}^{0.16666666666666666}} \]

      16. *-commutative 6.8%

          \[\leadsto \sqrt{x + 1} - \color{blue}{{x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

   6. Simplified 6.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} - {x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

   8. Applied egg-rr 4.2%

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

   9. Taylor expanded in x around 0 20.3%

      \[\leadsto \frac{\color{blue}{-1}}{-\sqrt{1 + \left(x + x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;\frac{x + \left(1 - x\right)}{\sqrt{x} + \left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sqrt{1 + \left(x + x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sqrt{1 + \left(x + x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.55)
   (- (+ 1.0 (* x (+ 0.5 (* x -0.125)))) (sqrt x))
   (/ -1.0 (- (sqrt (+ 1.0 (+ x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - sqrt(x);
	} else {
		tmp = -1.0 / -sqrt((1.0 + (x + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.55d0) then
        tmp = (1.0d0 + (x * (0.5d0 + (x * (-0.125d0))))) - sqrt(x)
    else
        tmp = (-1.0d0) / -sqrt((1.0d0 + (x + x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - Math.sqrt(x);
	} else {
		tmp = -1.0 / -Math.sqrt((1.0 + (x + x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - math.sqrt(x)
	else:
		tmp = -1.0 / -math.sqrt((1.0 + (x + x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * -0.125)))) - sqrt(x));
	else
		tmp = Float64(-1.0 / Float64(-sqrt(Float64(1.0 + Float64(x + x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.55)
		tmp = (1.0 + (x * (0.5 + (x * -0.125)))) - sqrt(x);
	else
		tmp = -1.0 / -sqrt((1.0 + (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.55], N[(N[(1.0 + N[(x * N[(0.5 + N[(x * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(-1.0 / (-N[Sqrt[N[(1.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 97.4%

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

      1. +-commutative 97.4%

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

      2. unpow2 97.4%

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

      3. associate-*r* 97.4%

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

      4. distribute-rgt-out 97.4%

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

      5. *-commutative 97.4%

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

   5. Simplified 97.4%

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

   if 1.55000000000000004 < x

   1. Initial program 4.9%

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

   4. Applied egg-rr 6.8%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. sub-neg 6.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(-\sqrt{x}\right)\right)} + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right) \]

      2. associate-+l+ 6.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)\right)} \]

      3. fma-undefine 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x} + \sqrt{x}\right)}\right) \]

      4. +-commutative 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\sqrt{x} + \left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x}\right)}\right) \]

      5. *-commutative 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\sqrt[3]{x} \cdot \left(-{x}^{0.16666666666666666}\right)}\right)\right) \]

      6. distribute-rgt-neg-out 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)}\right)\right) \]

      7. distribute-lft-neg-out 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}}\right)\right) \]

      8. associate-+r+ 6.8%

         \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\left(-\sqrt{x}\right) + \sqrt{x}\right) + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right)} \]

      9. +-commutative 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} + \left(-\sqrt{x}\right)\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      10. unsub-neg 6.8%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} - \sqrt{x}\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      11. +-inverses 6.8%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{0} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      12. +-commutative 6.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666} + 0\right)} \]

      13. +-rgt-identity 6.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}} \]

      14. distribute-lft-neg-out 6.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)} \]

      15. unsub-neg 6.8%

          \[\leadsto \color{blue}{\sqrt{x + 1} - \sqrt[3]{x} \cdot {x}^{0.16666666666666666}} \]

      16. *-commutative 6.8%

          \[\leadsto \sqrt{x + 1} - \color{blue}{{x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

   6. Simplified 6.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} - {x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

   8. Applied egg-rr 4.2%

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

   9. Taylor expanded in x around 0 20.3%

      \[\leadsto \frac{\color{blue}{-1}}{-\sqrt{1 + \left(x + x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;\left(1 + x \cdot \left(0.5 + x \cdot -0.125\right)\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sqrt{1 + \left(x + x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sqrt{1 + \left(x + x\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4)
   (- (+ 1.0 (* x 0.5)) (sqrt x))
   (/ -1.0 (- (sqrt (+ 1.0 (+ x x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	} else {
		tmp = -1.0 / -sqrt((1.0 + (x + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = (1.0d0 + (x * 0.5d0)) - sqrt(x)
    else
        tmp = (-1.0d0) / -sqrt((1.0d0 + (x + x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4) {
		tmp = (1.0 + (x * 0.5)) - Math.sqrt(x);
	} else {
		tmp = -1.0 / -Math.sqrt((1.0 + (x + x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4:
		tmp = (1.0 + (x * 0.5)) - math.sqrt(x)
	else:
		tmp = -1.0 / -math.sqrt((1.0 + (x + x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x));
	else
		tmp = Float64(-1.0 / Float64(-sqrt(Float64(1.0 + Float64(x + x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = (1.0 + (x * 0.5)) - sqrt(x);
	else
		tmp = -1.0 / -sqrt((1.0 + (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4], N[(N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(-1.0 / (-N[Sqrt[N[(1.0 + N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 96.9%

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

   if 1.3999999999999999 < x

   1. Initial program 4.9%

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

   4. Applied egg-rr 6.8%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)} \]
   5. Step-by-step derivation

      1. sub-neg 6.8%

         \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(-\sqrt{x}\right)\right)} + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right) \]

      2. associate-+l+ 6.8%

         \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)\right)} \]

      3. fma-undefine 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x} + \sqrt{x}\right)}\right) \]

      4. +-commutative 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\sqrt{x} + \left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x}\right)}\right) \]

      5. *-commutative 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\sqrt[3]{x} \cdot \left(-{x}^{0.16666666666666666}\right)}\right)\right) \]

      6. distribute-rgt-neg-out 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)}\right)\right) \]

      7. distribute-lft-neg-out 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}}\right)\right) \]

      8. associate-+r+ 6.8%

         \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\left(-\sqrt{x}\right) + \sqrt{x}\right) + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right)} \]

      9. +-commutative 6.8%

         \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} + \left(-\sqrt{x}\right)\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      10. unsub-neg 6.8%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} - \sqrt{x}\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      11. +-inverses 6.8%

          \[\leadsto \sqrt{x + 1} + \left(\color{blue}{0} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

      12. +-commutative 6.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666} + 0\right)} \]

      13. +-rgt-identity 6.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}} \]

      14. distribute-lft-neg-out 6.8%

          \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)} \]

      15. unsub-neg 6.8%

          \[\leadsto \color{blue}{\sqrt{x + 1} - \sqrt[3]{x} \cdot {x}^{0.16666666666666666}} \]

      16. *-commutative 6.8%

          \[\leadsto \sqrt{x + 1} - \color{blue}{{x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

   6. Simplified 6.8%

      \[\leadsto \color{blue}{\sqrt{1 + x} - {x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

   8. Applied egg-rr 4.2%

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

   9. Taylor expanded in x around 0 20.3%

      \[\leadsto \frac{\color{blue}{-1}}{-\sqrt{1 + \left(x + x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\left(1 + x \cdot 0.5\right) - \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-\sqrt{1 + \left(x + x\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(1 + x \cdot 0.5\right) - \sqrt{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (+ 1.0 (* x 0.5)) (sqrt x)))

C

double code(double x) {
	return (1.0 + (x * 0.5)) - sqrt(x);
}

Fortran

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

Java

public static double code(double x) {
	return (1.0 + (x * 0.5)) - Math.sqrt(x);
}

Python

def code(x):
	return (1.0 + (x * 0.5)) - math.sqrt(x)

Julia

function code(x)
	return Float64(Float64(1.0 + Float64(x * 0.5)) - sqrt(x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 + (x * 0.5)) - sqrt(x);
end

Wolfram

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

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in x around 0 55.0%

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

4. Final simplification 55.0%

   \[\leadsto \left(1 + x \cdot 0.5\right) - \sqrt{x} \]
5. Add Preprocessing

Alternative 10: 52.1% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + x \cdot 1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ 1.0 (* x 1.5))))

C

double code(double x) {
	return 1.0 / (1.0 + (x * 1.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 + (x * 1.5d0))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 + (x * 1.5));
}

Python

def code(x):
	return 1.0 / (1.0 + (x * 1.5))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 + Float64(x * 1.5)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 + (x * 1.5));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[(x * 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

   1. pow1/2 56.8%

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

   2. pow1 56.8%

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

   3. pow-to-exp 56.7%

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

   4. pow-exp 56.7%

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

   5. log1p-expm1-u 56.7%

      \[\leadsto e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(x + 1\right) \cdot 1\right)\right)} \cdot 0.5} - \sqrt{x} \]

   6. expm1-define 56.7%

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

   7. pow-exp 56.7%

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

   8. pow1 56.7%

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

   9. expm1-undefine 56.7%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(x + 1\right)\right)}\right) \cdot 0.5} - \sqrt{x} \]

   10. log1p-expm1-u 56.7%

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

   11. +-commutative 56.7%

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

   12. log1p-define 56.7%

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

4. Applied egg-rr 56.7%

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

5. Taylor expanded in x around 0 55.0%

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

   1. *-commutative 55.0%

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

7. Simplified 55.0%

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

9. Applied egg-rr 53.8%

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

    1. exp-prod 53.8%

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

    2. unpow-1 53.8%

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

    3. log1p-undefine 53.8%

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

    4. rem-exp-log 53.8%

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

    5. fma-undefine 53.8%

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

    6. *-commutative 53.8%

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

    7. distribute-lft1-in 53.8%

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

    8. metadata-eval 53.8%

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

11. Simplified 53.8%

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

12. Final simplification 53.8%

    \[\leadsto \frac{1}{1 + x \cdot 1.5} \]
13. Add Preprocessing

Alternative 11: 52.1% accurate, 41.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ -1.0 (- -1.0 x)))

C

double code(double x) {
	return -1.0 / (-1.0 - x);
}

Fortran

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

Java

public static double code(double x) {
	return -1.0 / (-1.0 - x);
}

Python

def code(x):
	return -1.0 / (-1.0 - x)

Julia

function code(x)
	return Float64(-1.0 / Float64(-1.0 - x))
end

MATLAB

function tmp = code(x)
	tmp = -1.0 / (-1.0 - x);
end

Wolfram

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

Derivation

1. Initial program 56.8%

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

4. Applied egg-rr 57.7%

   \[\leadsto \color{blue}{\left(\sqrt{x + 1} - \sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)} \]
5. Step-by-step derivation

   1. sub-neg 57.7%

      \[\leadsto \color{blue}{\left(\sqrt{x + 1} + \left(-\sqrt{x}\right)\right)} + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right) \]

   2. associate-+l+ 57.7%

      \[\leadsto \color{blue}{\sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \mathsf{fma}\left(-{x}^{0.16666666666666666}, \sqrt[3]{x}, \sqrt{x}\right)\right)} \]

   3. fma-undefine 57.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x} + \sqrt{x}\right)}\right) \]

   4. +-commutative 57.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \color{blue}{\left(\sqrt{x} + \left(-{x}^{0.16666666666666666}\right) \cdot \sqrt[3]{x}\right)}\right) \]

   5. *-commutative 57.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\sqrt[3]{x} \cdot \left(-{x}^{0.16666666666666666}\right)}\right)\right) \]

   6. distribute-rgt-neg-out 57.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)}\right)\right) \]

   7. distribute-lft-neg-out 57.7%

      \[\leadsto \sqrt{x + 1} + \left(\left(-\sqrt{x}\right) + \left(\sqrt{x} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}}\right)\right) \]

   8. associate-+r+ 57.7%

      \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(\left(-\sqrt{x}\right) + \sqrt{x}\right) + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right)} \]

   9. +-commutative 57.7%

      \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} + \left(-\sqrt{x}\right)\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

   10. unsub-neg 57.7%

       \[\leadsto \sqrt{x + 1} + \left(\color{blue}{\left(\sqrt{x} - \sqrt{x}\right)} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

   11. +-inverses 57.7%

       \[\leadsto \sqrt{x + 1} + \left(\color{blue}{0} + \left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}\right) \]

   12. +-commutative 57.7%

       \[\leadsto \sqrt{x + 1} + \color{blue}{\left(\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666} + 0\right)} \]

   13. +-rgt-identity 57.7%

       \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x}\right) \cdot {x}^{0.16666666666666666}} \]

   14. distribute-lft-neg-out 57.7%

       \[\leadsto \sqrt{x + 1} + \color{blue}{\left(-\sqrt[3]{x} \cdot {x}^{0.16666666666666666}\right)} \]

   15. unsub-neg 57.7%

       \[\leadsto \color{blue}{\sqrt{x + 1} - \sqrt[3]{x} \cdot {x}^{0.16666666666666666}} \]

   16. *-commutative 57.7%

       \[\leadsto \sqrt{x + 1} - \color{blue}{{x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

6. Simplified 57.7%

   \[\leadsto \color{blue}{\sqrt{1 + x} - {x}^{0.16666666666666666} \cdot \sqrt[3]{x}} \]

8. Applied egg-rr 52.5%

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

9. Taylor expanded in x around 0 59.8%

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

10. Taylor expanded in x around 0 53.8%

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

11. Final simplification 53.8%

    \[\leadsto \frac{-1}{-1 - x} \]
12. Add Preprocessing

Alternative 12: 52.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in x around 0 53.7%

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

4. Final simplification 53.7%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024039 
(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)))