2isqrt (example 3.6)

Percentage Accurate: 68.8% → 99.6%

Time: 13.2s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.8%

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

   1. frac-sub 68.8%

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

   2. *-un-lft-identity 68.8%

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

   3. *-rgt-identity 68.8%

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

   4. flip-- 68.8%

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

   5. associate-/l/ 68.8%

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

   6. add-sqr-sqrt 68.6%

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

   7. +-commutative 68.6%

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

   8. add-sqr-sqrt 69.3%

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

   9. associate--l+ 99.2%

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

   10. sqrt-unprod 92.6%

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

   11. +-commutative 92.6%

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

   12. distribute-rgt-in 92.6%

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

   13. *-un-lft-identity 92.6%

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

   14. pow2 92.6%

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

   15. +-commutative 92.6%

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

   16. +-commutative 92.6%

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

3. Applied egg-rr 92.6%

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

   1. +-inverses 92.6%

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

   2. metadata-eval 92.6%

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

   3. +-commutative 92.6%

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

5. Simplified 92.6%

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

   1. inv-pow 92.6%

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

   2. add-sqr-sqrt 92.3%

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

   3. metadata-eval 92.3%

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

   4. unpow-prod-down 92.0%

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

7. Applied egg-rr 98.8%

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

   1. pow-sqr 99.0%

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

   2. metadata-eval 99.0%

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

9. Simplified 99.0%

   \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(x, \sqrt{x}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}\right)}^{-2}} \]
10. Step-by-step derivation

    1. sqrt-pow2 99.4%

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

    2. metadata-eval 99.4%

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

    3. unpow-1 99.4%

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

    4. *-commutative 99.4%

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

    5. associate-/r* 99.6%

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

    6. +-commutative 99.6%

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

11. Applied egg-rr 99.6%

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

12. Final simplification 99.6%

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

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt{x} + t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 2e-23)
     (/ 1.0 (* x (+ (sqrt x) t_0)))
     (- (pow x -0.5) (/ 1.0 t_0)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 2e-23) {
		tmp = 1.0 / (x * (sqrt(x) + t_0));
	} else {
		tmp = pow(x, -0.5) - (1.0 / 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))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 2d-23) then
        tmp = 1.0d0 / (x * (sqrt(x) + t_0))
    else
        tmp = (x ** (-0.5d0)) - (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 2e-23) {
		tmp = 1.0 / (x * (Math.sqrt(x) + t_0));
	} else {
		tmp = Math.pow(x, -0.5) - (1.0 / t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 2e-23:
		tmp = 1.0 / (x * (math.sqrt(x) + t_0))
	else:
		tmp = math.pow(x, -0.5) - (1.0 / t_0)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 2e-23)
		tmp = Float64(1.0 / Float64(x * Float64(sqrt(x) + t_0)));
	else
		tmp = Float64((x ^ -0.5) - Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 2e-23)
		tmp = 1.0 / (x * (sqrt(x) + t_0));
	else
		tmp = (x ^ -0.5) - (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 2e-23], N[(1.0 / N[(x * N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1.99999999999999992e-23

   1. Initial program 35.8%

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

      1. frac-sub 35.8%

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

      2. *-un-lft-identity 35.8%

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

      3. *-rgt-identity 35.8%

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

      4. flip-- 35.8%

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

      5. associate-/l/ 35.8%

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

      6. add-sqr-sqrt 35.2%

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

      7. +-commutative 35.2%

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

      8. add-sqr-sqrt 36.4%

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

      9. associate--l+ 98.7%

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

      10. sqrt-unprod 84.9%

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

      11. +-commutative 84.9%

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

      12. distribute-rgt-in 84.9%

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

      13. *-un-lft-identity 84.9%

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

      14. pow2 84.9%

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

      15. +-commutative 84.9%

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

      16. +-commutative 84.9%

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

   3. Applied egg-rr 84.9%

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

      1. +-inverses 84.9%

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

      2. metadata-eval 84.9%

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

      3. +-commutative 84.9%

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

   5. Simplified 84.9%

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

   6. Taylor expanded in x around inf 99.0%

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

   if 1.99999999999999992e-23 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.3%

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

      1. expm1-log1p-u 89.5%

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

      2. expm1-udef 89.5%

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

      3. pow1/2 89.5%

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

      4. pow-flip 89.5%

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

      5. metadata-eval 89.5%

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

   3. Applied egg-rr 91.8%

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

      1. expm1-def 89.5%

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

      2. expm1-log1p 96.9%

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{\sqrt{x + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt{x} + \sqrt{1 + x}\right)}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{\sqrt{1 + x}}\\ \end{array} \]

Alternative 3: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{x} + t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 2e-23)
     (/ (/ 1.0 x) (+ (sqrt x) t_0))
     (- (pow x -0.5) (/ 1.0 t_0)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 2e-23) {
		tmp = (1.0 / x) / (sqrt(x) + t_0);
	} else {
		tmp = pow(x, -0.5) - (1.0 / 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))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 2d-23) then
        tmp = (1.0d0 / x) / (sqrt(x) + t_0)
    else
        tmp = (x ** (-0.5d0)) - (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 2e-23) {
		tmp = (1.0 / x) / (Math.sqrt(x) + t_0);
	} else {
		tmp = Math.pow(x, -0.5) - (1.0 / t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 2e-23:
		tmp = (1.0 / x) / (math.sqrt(x) + t_0)
	else:
		tmp = math.pow(x, -0.5) - (1.0 / t_0)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 2e-23)
		tmp = Float64(Float64(1.0 / x) / Float64(sqrt(x) + t_0));
	else
		tmp = Float64((x ^ -0.5) - Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 2e-23)
		tmp = (1.0 / x) / (sqrt(x) + t_0);
	else
		tmp = (x ^ -0.5) - (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 2e-23], N[(N[(1.0 / x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1.99999999999999992e-23

   1. Initial program 35.8%

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

      1. frac-sub 35.8%

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

      2. *-un-lft-identity 35.8%

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

      3. *-rgt-identity 35.8%

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

      4. flip-- 35.8%

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

      5. associate-/l/ 35.8%

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

      6. add-sqr-sqrt 35.2%

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

      7. +-commutative 35.2%

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

      8. add-sqr-sqrt 36.4%

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

      9. associate--l+ 98.7%

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

      10. sqrt-unprod 84.9%

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

      11. +-commutative 84.9%

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

      12. distribute-rgt-in 84.9%

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

      13. *-un-lft-identity 84.9%

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

      14. pow2 84.9%

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

      15. +-commutative 84.9%

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

      16. +-commutative 84.9%

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

   3. Applied egg-rr 84.9%

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

      1. +-inverses 84.9%

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

      2. metadata-eval 84.9%

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

      3. +-commutative 84.9%

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

   5. Simplified 84.9%

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

      1. expm1-log1p-u 84.9%

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

      2. expm1-udef 35.6%

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

      3. associate-/r* 35.6%

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

      4. +-commutative 35.6%

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

      5. unpow2 35.6%

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

      6. add-sqr-sqrt 35.6%

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

      7. hypot-def 35.6%

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

      8. +-commutative 35.6%

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

      9. +-commutative 35.6%

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

   7. Applied egg-rr 35.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}}{\sqrt{x} + \sqrt{x + 1}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 99.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}}{\sqrt{x} + \sqrt{x + 1}}\right)\right)} \]

      2. expm1-log1p 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in x around inf 99.6%

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

   if 1.99999999999999992e-23 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.3%

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

      1. expm1-log1p-u 89.5%

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

      2. expm1-udef 89.5%

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

      3. pow1/2 89.5%

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

      4. pow-flip 89.5%

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

      5. metadata-eval 89.5%

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

   3. Applied egg-rr 91.8%

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

      1. expm1-def 89.5%

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

      2. expm1-log1p 96.9%

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{\sqrt{x + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{x}}{\sqrt{x} + \sqrt{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{\sqrt{1 + x}}\\ \end{array} \]

Alternative 4: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 2e-23)
     (/ (/ 1.0 (+ (sqrt x) t_0)) x)
     (- (pow x -0.5) (/ 1.0 t_0)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 2e-23) {
		tmp = (1.0 / (sqrt(x) + t_0)) / x;
	} else {
		tmp = pow(x, -0.5) - (1.0 / 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))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 2d-23) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) / x
    else
        tmp = (x ** (-0.5d0)) - (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 2e-23) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) / x;
	} else {
		tmp = Math.pow(x, -0.5) - (1.0 / t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 2e-23:
		tmp = (1.0 / (math.sqrt(x) + t_0)) / x
	else:
		tmp = math.pow(x, -0.5) - (1.0 / t_0)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 2e-23)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) / x);
	else
		tmp = Float64((x ^ -0.5) - Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 2e-23)
		tmp = (1.0 / (sqrt(x) + t_0)) / x;
	else
		tmp = (x ^ -0.5) - (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 2e-23], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1.99999999999999992e-23

   1. Initial program 35.8%

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

      1. frac-sub 35.8%

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

      2. *-un-lft-identity 35.8%

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

      3. *-rgt-identity 35.8%

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

      4. flip-- 35.8%

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

      5. associate-/l/ 35.8%

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

      6. add-sqr-sqrt 35.2%

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

      7. +-commutative 35.2%

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

      8. add-sqr-sqrt 36.4%

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

      9. associate--l+ 98.7%

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

      10. sqrt-unprod 84.9%

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

      11. +-commutative 84.9%

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

      12. distribute-rgt-in 84.9%

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

      13. *-un-lft-identity 84.9%

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

      14. pow2 84.9%

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

      15. +-commutative 84.9%

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

      16. +-commutative 84.9%

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

   3. Applied egg-rr 84.9%

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

      1. +-inverses 84.9%

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

      2. metadata-eval 84.9%

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

      3. +-commutative 84.9%

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

   5. Simplified 84.9%

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

      1. inv-pow 84.9%

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

      2. add-sqr-sqrt 84.8%

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

      3. metadata-eval 84.8%

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

      4. unpow-prod-down 84.6%

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

   7. Applied egg-rr 98.6%

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

      1. pow-sqr 98.9%

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

      2. metadata-eval 98.9%

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

   9. Simplified 98.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(x, \sqrt{x}\right) \cdot \left(\sqrt{x} + \sqrt{x + 1}\right)}\right)}^{-2}} \]
   10. Step-by-step derivation

       1. sqrt-pow2 99.0%

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

       2. metadata-eval 99.0%

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

       3. unpow-1 99.0%

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

       4. *-commutative 99.0%

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

       5. associate-/r* 99.6%

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

       6. +-commutative 99.6%

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

   11. Applied egg-rr 99.6%

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

   12. Taylor expanded in x around inf 99.6%

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

   if 1.99999999999999992e-23 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.3%

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

      1. expm1-log1p-u 89.5%

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

      2. expm1-udef 89.5%

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

      3. pow1/2 89.5%

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

      4. pow-flip 89.5%

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

      5. metadata-eval 89.5%

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

   3. Applied egg-rr 91.8%

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

      1. expm1-def 89.5%

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

      2. expm1-log1p 96.9%

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{\sqrt{x + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{\sqrt{1 + x}}\\ \end{array} \]

Alternative 5: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 + x}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{t_0} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ 1.0 x))))
   (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 t_0)) 2e-23)
     (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))
     (- (pow x -0.5) (/ 1.0 t_0)))))

C

double code(double x) {
	double t_0 = sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 2e-23) {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	} else {
		tmp = pow(x, -0.5) - (1.0 / 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))
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / t_0)) <= 2d-23) then
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    else
        tmp = (x ** (-0.5d0)) - (1.0d0 / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((1.0 + x));
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / t_0)) <= 2e-23) {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	} else {
		tmp = Math.pow(x, -0.5) - (1.0 / t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((1.0 + x))
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / t_0)) <= 2e-23:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	else:
		tmp = math.pow(x, -0.5) - (1.0 / t_0)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(1.0 + x))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / t_0)) <= 2e-23)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	else
		tmp = Float64((x ^ -0.5) - Float64(1.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((1.0 + x));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / t_0)) <= 2e-23)
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	else
		tmp = (x ^ -0.5) - (1.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 2e-23], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1.99999999999999992e-23

   1. Initial program 35.8%

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

      1. flip-- 35.8%

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

      2. flip-- 35.8%

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

      3. associate-/l/ 7.4%

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

   3. Applied egg-rr 7.4%

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

   4. Taylor expanded in x around inf 59.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]

   if 1.99999999999999992e-23 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.3%

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

      1. expm1-log1p-u 89.5%

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

      2. expm1-udef 89.5%

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

      3. pow1/2 89.5%

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

      4. pow-flip 89.5%

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

      5. metadata-eval 89.5%

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

   3. Applied egg-rr 91.8%

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

      1. expm1-def 89.5%

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

      2. expm1-log1p 96.9%

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

   5. Simplified 99.6%

      \[\leadsto \color{blue}{{x}^{-0.5}} - \frac{1}{\sqrt{x + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-23}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{\sqrt{1 + x}}\\ \end{array} \]

Alternative 6: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 118000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 118000000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 118000000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 118000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 118000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 118000000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 118000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 118000000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 118000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.18e8

   1. Initial program 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 99.1%

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

      3. associate--r- 99.1%

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

      4. pow1/2 99.1%

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

      5. pow-flip 99.4%

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

      6. metadata-eval 99.4%

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

      7. pow1/2 99.4%

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

      8. pow-flip 99.4%

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

      9. +-commutative 99.4%

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

      10. metadata-eval 99.4%

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

   3. Applied egg-rr 99.4%

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

      1. associate-+l- 99.4%

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

      2. expm1-def 99.6%

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

      3. expm1-log1p 99.6%

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

   5. Simplified 99.6%

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

   if 1.18e8 < x

   1. Initial program 35.8%

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

      1. flip-- 35.8%

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

      2. flip-- 35.8%

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

      3. associate-/l/ 7.4%

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

   3. Applied egg-rr 7.4%

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

   4. Taylor expanded in x around inf 59.0%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 118000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 7: 81.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.68:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.68)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.68) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.68d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.68) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.68:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.68)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.68)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.68], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.67999999999999994

   1. Initial program 99.7%

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

      1. expm1-log1p-u 91.5%

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

      2. expm1-udef 91.5%

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

      3. pow1/2 91.5%

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

      4. pow-flip 91.5%

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

      5. metadata-eval 91.5%

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

   3. Applied egg-rr 92.3%

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

      1. expm1-def 91.5%

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

      2. expm1-log1p 99.1%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   if 1.67999999999999994 < x

   1. Initial program 36.9%

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

      1. flip-- 36.9%

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

      2. flip-- 36.9%

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

      3. associate-/l/ 9.2%

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

   3. Applied egg-rr 9.2%

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

   4. Taylor expanded in x around inf 58.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.68:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]

Alternative 8: 68.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot 0.5\\ \mathbf{if}\;x \leq 1.32:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt{x} + t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x 0.5))))
   (if (<= x 1.32)
     (+ (pow x -0.5) (/ -1.0 t_0))
     (/ 1.0 (* x (+ (sqrt x) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 + (x * 0.5);
	double tmp;
	if (x <= 1.32) {
		tmp = pow(x, -0.5) + (-1.0 / t_0);
	} else {
		tmp = 1.0 / (x * (sqrt(x) + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * 0.5d0)
    if (x <= 1.32d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / t_0)
    else
        tmp = 1.0d0 / (x * (sqrt(x) + t_0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 + (x * 0.5);
	double tmp;
	if (x <= 1.32) {
		tmp = Math.pow(x, -0.5) + (-1.0 / t_0);
	} else {
		tmp = 1.0 / (x * (Math.sqrt(x) + t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 + (x * 0.5)
	tmp = 0
	if x <= 1.32:
		tmp = math.pow(x, -0.5) + (-1.0 / t_0)
	else:
		tmp = 1.0 / (x * (math.sqrt(x) + t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * 0.5))
	tmp = 0.0
	if (x <= 1.32)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / t_0));
	else
		tmp = Float64(1.0 / Float64(x * Float64(sqrt(x) + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 + (x * 0.5);
	tmp = 0.0;
	if (x <= 1.32)
		tmp = (x ^ -0.5) + (-1.0 / t_0);
	else
		tmp = 1.0 / (x * (sqrt(x) + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.32], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x * N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.32000000000000006

   1. Initial program 99.7%

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

      1. expm1-log1p-u 91.5%

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

      2. expm1-udef 91.5%

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

      3. pow1/2 91.5%

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

      4. pow-flip 91.5%

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

      5. metadata-eval 91.5%

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

   3. Applied egg-rr 92.3%

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

      1. expm1-def 91.5%

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

      2. expm1-log1p 99.1%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

   8. Simplified 99.7%

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

   if 1.32000000000000006 < x

   1. Initial program 36.9%

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

      1. frac-sub 36.9%

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

      2. *-un-lft-identity 36.9%

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

      3. *-rgt-identity 36.9%

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

      4. flip-- 36.9%

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

      5. associate-/l/ 36.9%

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

      6. add-sqr-sqrt 36.5%

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

      7. +-commutative 36.5%

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

      8. add-sqr-sqrt 37.9%

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

      9. associate--l+ 98.8%

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

      10. sqrt-unprod 85.3%

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

      11. +-commutative 85.3%

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

      12. distribute-rgt-in 85.3%

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

      13. *-un-lft-identity 85.3%

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

      14. pow2 85.3%

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

      15. +-commutative 85.3%

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

      16. +-commutative 85.3%

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

   3. Applied egg-rr 85.3%

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

      1. +-inverses 85.3%

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

      2. metadata-eval 85.3%

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

      3. +-commutative 85.3%

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

   5. Simplified 85.3%

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

   6. Taylor expanded in x around inf 97.6%

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

   7. Taylor expanded in x around 0 36.7%

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

      1. *-commutative 36.7%

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

   9. Simplified 36.7%

      \[\leadsto \frac{1}{x \cdot \left(\color{blue}{\left(1 + x \cdot 0.5\right)} + \sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.32:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \left(\sqrt{x} + \left(1 + x \cdot 0.5\right)\right)}\\ \end{array} \]

Alternative 9: 53.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.65) (+ (pow x -0.5) (- -1.0 (* x -0.5))) (/ 1.0 x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = pow(x, -0.5) + (-1.0 - (x * -0.5));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.65d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) - (x * (-0.5d0)))
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.65) {
		tmp = Math.pow(x, -0.5) + (-1.0 - (x * -0.5));
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.65:
		tmp = math.pow(x, -0.5) + (-1.0 - (x * -0.5))
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.65)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 - Float64(x * -0.5)));
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.65)
		tmp = (x ^ -0.5) + (-1.0 - (x * -0.5));
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.65], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 - N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6499999999999999

   1. Initial program 99.7%

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

      1. expm1-log1p-u 91.5%

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

      2. expm1-udef 91.5%

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

      3. pow1/2 91.5%

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

      4. pow-flip 91.5%

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

      5. metadata-eval 91.5%

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

   3. Applied egg-rr 92.3%

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

      1. expm1-def 91.5%

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

      2. expm1-log1p 99.1%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 99.7%

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

   if 1.6499999999999999 < x

   1. Initial program 36.9%

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

      1. frac-sub 36.9%

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

      2. *-un-lft-identity 36.9%

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

      3. *-rgt-identity 36.9%

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

      4. flip-- 36.9%

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

      5. associate-/l/ 36.9%

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

      6. add-sqr-sqrt 36.5%

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

      7. +-commutative 36.5%

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

      8. add-sqr-sqrt 37.9%

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

      9. associate--l+ 98.8%

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

      10. sqrt-unprod 85.3%

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

      11. +-commutative 85.3%

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

      12. distribute-rgt-in 85.3%

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

      13. *-un-lft-identity 85.3%

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

      14. pow2 85.3%

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

      15. +-commutative 85.3%

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

      16. +-commutative 85.3%

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

   3. Applied egg-rr 85.3%

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

      1. +-inverses 85.3%

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

      2. metadata-eval 85.3%

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

      3. +-commutative 85.3%

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

   5. Simplified 85.3%

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

   6. Taylor expanded in x around inf 97.6%

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

   7. Taylor expanded in x around 0 7.5%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 10: 52.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	return 1.0 / (x * (1.0 + pow(x, -0.5)));
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / (x * (1.0 + math.pow(x, -0.5)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.8%

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

   1. flip-- 68.7%

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

   2. flip-- 41.5%

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

   3. associate-/l/ 27.0%

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

3. Applied egg-rr 27.0%

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

4. Taylor expanded in x around 0 53.4%

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

5. Final simplification 53.4%

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

Alternative 11: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.82:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.82) (+ (pow x -0.5) -1.0) (/ 1.0 x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.82) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.82d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 1.0d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.82) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.82:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 1.0 / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.82)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(1.0 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.82)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 1.0 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.82], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.819999999999999951

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 98.9%

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

      1. expm1-log1p-u 91.5%

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

      2. expm1-udef 91.5%

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

      3. pow1/2 91.5%

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

      4. pow-flip 91.5%

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

      5. metadata-eval 91.5%

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

   4. Applied egg-rr 91.5%

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

      1. expm1-def 91.5%

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

      2. expm1-log1p 99.1%

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

   6. Simplified 99.1%

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

   if 0.819999999999999951 < x

   1. Initial program 36.9%

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

      1. frac-sub 36.9%

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

      2. *-un-lft-identity 36.9%

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

      3. *-rgt-identity 36.9%

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

      4. flip-- 36.9%

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

      5. associate-/l/ 36.9%

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

      6. add-sqr-sqrt 36.5%

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

      7. +-commutative 36.5%

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

      8. add-sqr-sqrt 37.9%

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

      9. associate--l+ 98.8%

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

      10. sqrt-unprod 85.3%

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

      11. +-commutative 85.3%

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

      12. distribute-rgt-in 85.3%

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

      13. *-un-lft-identity 85.3%

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

      14. pow2 85.3%

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

      15. +-commutative 85.3%

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

      16. +-commutative 85.3%

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

   3. Applied egg-rr 85.3%

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

      1. +-inverses 85.3%

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

      2. metadata-eval 85.3%

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

      3. +-commutative 85.3%

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

   5. Simplified 85.3%

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

   6. Taylor expanded in x around inf 97.6%

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

   7. Taylor expanded in x around 0 7.5%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.82:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]

Alternative 12: 50.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow x -0.5))

C

double code(double x) {
	return pow(x, -0.5);
}

Fortran

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

Java

public static double code(double x) {
	return Math.pow(x, -0.5);
}

Python

def code(x):
	return math.pow(x, -0.5)

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

function tmp = code(x)
	tmp = x ^ -0.5;
end

Wolfram

code[x_] := N[Power[x, -0.5], $MachinePrecision]

Derivation

1. Initial program 68.8%

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

   1. inv-pow 68.8%

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

   2. add-sqr-sqrt 61.1%

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

   3. metadata-eval 61.1%

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

   4. unpow-prod-down 57.8%

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

   5. pow1/2 57.8%

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

   6. sqrt-pow1 59.4%

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

   7. metadata-eval 59.4%

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

   8. metadata-eval 59.4%

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

   9. pow1/2 59.4%

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

   10. sqrt-pow1 57.9%

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

   11. metadata-eval 57.9%

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

   12. metadata-eval 57.9%

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

3. Applied egg-rr 57.9%

   \[\leadsto \color{blue}{{\left({x}^{0.25}\right)}^{-1} \cdot {\left({x}^{0.25}\right)}^{-1}} - \frac{1}{\sqrt{x + 1}} \]
4. Step-by-step derivation

   1. pow-sqr 60.2%

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

   2. metadata-eval 60.2%

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

5. Simplified 60.2%

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

6. Taylor expanded in x around inf 52.0%

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

   1. unpow1/2 52.0%

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

   2. rem-exp-log 48.3%

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

   3. exp-neg 48.3%

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

   4. exp-prod 48.3%

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

   5. *-commutative 48.3%

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

   6. neg-mul-1 48.3%

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

   7. associate-*r* 48.3%

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

   8. metadata-eval 48.3%

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

   9. *-commutative 48.3%

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

   10. exp-to-pow 52.1%

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

8. Simplified 52.1%

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

9. Final simplification 52.1%

   \[\leadsto {x}^{-0.5} \]

Alternative 13: 7.4% accurate, 69.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.8%

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

   1. frac-sub 68.8%

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

   2. *-un-lft-identity 68.8%

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

   3. *-rgt-identity 68.8%

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

   4. flip-- 68.8%

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

   5. associate-/l/ 68.8%

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

   6. add-sqr-sqrt 68.6%

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

   7. +-commutative 68.6%

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

   8. add-sqr-sqrt 69.3%

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

   9. associate--l+ 99.2%

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

   10. sqrt-unprod 92.6%

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

   11. +-commutative 92.6%

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

   12. distribute-rgt-in 92.6%

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

   13. *-un-lft-identity 92.6%

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

   14. pow2 92.6%

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

   15. +-commutative 92.6%

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

   16. +-commutative 92.6%

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

3. Applied egg-rr 92.6%

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

   1. +-inverses 92.6%

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

   2. metadata-eval 92.6%

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

   3. +-commutative 92.6%

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

5. Simplified 92.6%

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

6. Taylor expanded in x around inf 51.4%

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

7. Taylor expanded in x around 0 7.1%

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

8. Final simplification 7.1%

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

Developer target: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023305 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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