Main:bigenough3 from C

Percentage Accurate: 52.6% → 99.5%

Time: 9.3s

Alternatives: 5

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sqrt((1.0 + x)) - sqrt(x);
	double tmp;
	if (t_0 <= 2e-6) {
		tmp = pow(x, -0.5) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((1.0d0 + x)) - sqrt(x)
    if (t_0 <= 2d-6) then
        tmp = (x ** (-0.5d0)) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 2e-6) {
		tmp = Math.pow(x, -0.5) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 2e-6:
		tmp = math.pow(x, -0.5) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(1.0 + x)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 2e-6)
		tmp = Float64((x ^ -0.5) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 2e-6)
		tmp = (x ^ -0.5) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-6], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 4.3%

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

      1. flip3-- 2.9%

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

      2. sqrt-pow2 3.2%

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

      3. metadata-eval 3.2%

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

      4. sqrt-pow2 2.9%

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

      5. metadata-eval 2.9%

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

      6. add-sqr-sqrt 2.9%

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

      7. add-sqr-sqrt 2.9%

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

      8. associate-+r+ 2.9%

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

      9. sqrt-unprod 2.9%

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

      10. add-sqr-sqrt 2.9%

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

      11. pow2 2.9%

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

      12. metadata-eval 2.9%

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

      13. add-sqr-sqrt 2.9%

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

      14. pow1 2.9%

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

      15. pow-plus 2.9%

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

      16. pow-prod-down 2.9%

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

   4. Applied egg-rr 2.9%

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

   5. Taylor expanded in x around inf 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. expm1-log1p-u 99.6%

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

      2. expm1-udef 6.4%

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

      3. inv-pow 6.4%

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

      4. sqrt-pow1 6.4%

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

      5. metadata-eval 6.4%

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

   9. Applied egg-rr 6.4%

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

       1. expm1-def 99.8%

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

       2. expm1-log1p 99.8%

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

   11. Simplified 99.8%

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

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

   1. Initial program 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

   1. flip-- 49.0%

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

   2. div-inv 49.0%

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

   3. add-sqr-sqrt 49.3%

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

   4. add-sqr-sqrt 49.6%

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

   5. associate--l+ 49.6%

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

4. Applied egg-rr 49.6%

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

   1. +-commutative 49.6%

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

   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%

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

Alternative 3: 97.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.6) (- 1.0 x) (* (pow x -0.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = 1.0 - x;
	} else {
		tmp = pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.6d0) then
        tmp = 1.0d0 - x
    else
        tmp = (x ** (-0.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.6) {
		tmp = 1.0 - x;
	} else {
		tmp = Math.pow(x, -0.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.6:
		tmp = 1.0 - x
	else:
		tmp = math.pow(x, -0.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.6)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64((x ^ -0.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.6)
		tmp = 1.0 - x;
	else
		tmp = (x ^ -0.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.6], N[(1.0 - x), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.599999999999999978

   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%

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

   8. Applied egg-rr 94.8%

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

   9. Taylor expanded in x around 0 94.8%

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

       1. neg-mul-1 94.8%

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

       2. unsub-neg 94.8%

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

   11. Simplified 94.8%

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

   if 0.599999999999999978 < x

   1. Initial program 4.9%

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

      1. flip3-- 3.5%

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

      2. sqrt-pow2 3.8%

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

      3. metadata-eval 3.8%

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

      4. sqrt-pow2 3.5%

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

      5. metadata-eval 3.5%

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

      6. add-sqr-sqrt 3.5%

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

      7. add-sqr-sqrt 3.5%

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

      8. associate-+r+ 3.5%

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

      9. sqrt-unprod 3.5%

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

      10. add-sqr-sqrt 3.5%

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

      11. pow2 3.5%

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

      12. metadata-eval 3.5%

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

      13. add-sqr-sqrt 3.5%

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

      14. pow1 3.5%

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

      15. pow-plus 3.5%

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

      16. pow-prod-down 3.5%

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

   4. Applied egg-rr 3.5%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

      1. expm1-log1p-u 99.2%

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

      2. expm1-udef 6.7%

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

      3. inv-pow 6.7%

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

      4. sqrt-pow1 6.7%

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

      5. metadata-eval 6.7%

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

   9. Applied egg-rr 6.7%

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

       1. expm1-def 99.3%

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

       2. expm1-log1p 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.6:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.8% 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 49.1%

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

   1. flip-- 49.0%

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

   2. div-inv 49.0%

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

   3. add-sqr-sqrt 49.3%

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

   4. add-sqr-sqrt 49.6%

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

   5. associate--l+ 49.6%

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

4. Applied egg-rr 49.6%

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

   1. +-commutative 49.6%

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

   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%

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

8. Applied egg-rr 54.9%

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

9. Taylor expanded in x around 0 47.7%

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

    1. +-commutative 47.7%

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

11. Simplified 47.7%

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

12. Final simplification 47.7%

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

Alternative 5: 50.8% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in x around 0 47.6%

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

4. Final simplification 47.6%

   \[\leadsto 1 \]
5. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024019 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

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

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