2isqrt (example 3.6)

Percentage Accurate: 68.8% → 99.6%

Time: 13.9s

Alternatives: 14

Speedup: 1.9×

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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + t_0}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 1e-9)
     (/ (/ 1.0 (+ (sqrt x) t_0)) (+ x 0.5))
     (- (pow x -0.5) (pow (+ x 1.0) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 1e-9) {
		tmp = (1.0 / (sqrt(x) + t_0)) / (x + 0.5);
	} else {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 1d-9) then
        tmp = (1.0d0 / (sqrt(x) + t_0)) / (x + 0.5d0)
    else
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 1e-9) {
		tmp = (1.0 / (Math.sqrt(x) + t_0)) / (x + 0.5);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 1e-9:
		tmp = (1.0 / (math.sqrt(x) + t_0)) / (x + 0.5)
	else:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 1e-9)
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + t_0)) / Float64(x + 0.5));
	else
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 1e-9)
		tmp = (1.0 / (sqrt(x) + t_0)) / (x + 0.5);
	else
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 1e-9], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1.00000000000000006e-9

   1. Initial program 35.7%

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

      1. frac-sub 35.7%

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

      2. *-un-lft-identity 35.7%

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

      3. +-commutative 35.7%

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

      4. *-rgt-identity 35.7%

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

      5. sqrt-unprod 35.7%

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

      6. +-commutative 35.7%

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

   4. Applied egg-rr 35.7%

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

   5. Taylor expanded in x around inf 35.7%

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

      1. flip-- 36.1%

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

      2. div-inv 36.1%

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

      3. add-sqr-sqrt 20.1%

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

      4. add-sqr-sqrt 37.7%

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

      5. associate--l+ 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. +-inverses 99.5%

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

      2. metadata-eval 99.5%

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

      3. *-lft-identity 99.5%

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

      4. +-commutative 99.5%

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

   9. Simplified 99.5%

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

   if 1.00000000000000006e-9 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-+l- 99.8%

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

      2. expm1-log1p 99.8%

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

      3. expm1-def 99.7%

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

      4. associate--l- 99.7%

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

      5. fma-udef 99.7%

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

      6. distribute-lft1-in 99.7%

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

      7. metadata-eval 99.7%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 99.7%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 99.7%

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

      10. expm1-def 99.8%

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

      11. expm1-log1p 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \leq 10^{-9}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{x + 1}}}{x + 0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{t_0} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- (/ 1.0 (sqrt x)) (/ 1.0 t_0)) 2e-14)
     (/ (/ 0.5 x) t_0)
     (- (pow x -0.5) (pow (+ x 1.0) -0.5)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 2e-14) {
		tmp = (0.5 / x) / t_0;
	} else {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0))
    if (((1.0d0 / sqrt(x)) - (1.0d0 / t_0)) <= 2d-14) then
        tmp = (0.5d0 / x) / t_0
    else
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if (((1.0 / Math.sqrt(x)) - (1.0 / t_0)) <= 2e-14) {
		tmp = (0.5 / x) / t_0;
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if ((1.0 / math.sqrt(x)) - (1.0 / t_0)) <= 2e-14:
		tmp = (0.5 / x) / t_0
	else:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	return tmp

Julia

function code(x)
	t_0 = sqrt(Float64(x + 1.0))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / t_0)) <= 2e-14)
		tmp = Float64(Float64(0.5 / x) / t_0);
	else
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0));
	tmp = 0.0;
	if (((1.0 / sqrt(x)) - (1.0 / t_0)) <= 2e-14)
		tmp = (0.5 / x) / t_0;
	else
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], 2e-14], N[(N[(0.5 / x), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 2e-14

   1. Initial program 35.4%

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

      1. frac-sub 35.4%

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

      2. associate-/r* 35.4%

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

      3. *-un-lft-identity 35.4%

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

      4. +-commutative 35.4%

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

      5. *-rgt-identity 35.4%

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

      6. +-commutative 35.4%

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

   4. Applied egg-rr 35.4%

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

      1. flip-- 35.9%

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

      2. div-inv 35.9%

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

      3. add-sqr-sqrt 19.6%

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

      4. add-sqr-sqrt 37.4%

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

      5. associate--l+ 99.6%

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

   6. Applied egg-rr 99.3%

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

      1. +-inverses 99.6%

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

      2. metadata-eval 99.6%

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

      3. *-lft-identity 99.6%

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

      4. +-commutative 99.6%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around inf 99.1%

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

   if 2e-14 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.2%

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

      1. *-un-lft-identity 99.2%

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

      2. clear-num 99.2%

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

      3. associate-/r/ 99.2%

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

      4. prod-diff 99.2%

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

      5. *-un-lft-identity 99.2%

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

      6. fma-neg 99.2%

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

      7. *-un-lft-identity 99.2%

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

      8. pow1/2 99.2%

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

      9. pow-flip 99.6%

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

      10. metadata-eval 99.6%

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

      11. pow1/2 99.6%

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

      12. pow-flip 99.6%

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

      13. +-commutative 99.6%

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

      14. metadata-eval 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. associate-+l- 99.6%

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

      2. expm1-log1p 99.6%

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

      3. expm1-def 99.4%

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

      4. associate--l- 99.4%

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

      5. fma-udef 99.4%

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

      6. distribute-lft1-in 99.4%

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

      7. metadata-eval 99.4%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 99.4%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 99.4%

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

      10. expm1-def 99.6%

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

      11. expm1-log1p 99.6%

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

   6. Simplified 99.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x)
	return Float64((Float64(x + 1.0) ^ -0.5) / Float64(x + hypot(x, sqrt(x))))
end

MATLAB

function tmp = code(x)
	tmp = ((x + 1.0) ^ -0.5) / (x + hypot(x, sqrt(x)));
end

Wolfram

code[x_] := N[(N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision] / N[(x + N[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.3%

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

   1. frac-sub 68.3%

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

   2. associate-/r* 68.3%

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

   3. *-un-lft-identity 68.3%

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

   4. +-commutative 68.3%

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

   5. *-rgt-identity 68.3%

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

   6. +-commutative 68.3%

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

4. Applied egg-rr 68.3%

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

   1. flip-- 21.6%

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

   2. div-inv 21.6%

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

   3. add-sqr-sqrt 13.8%

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

   4. add-sqr-sqrt 22.4%

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

   5. associate--l+ 52.6%

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

6. Applied egg-rr 99.4%

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

   1. +-inverses 52.6%

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

   2. metadata-eval 52.6%

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

   3. *-lft-identity 52.6%

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

   4. +-commutative 52.6%

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

8. Simplified 99.4%

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

   1. expm1-log1p-u 96.0%

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

   2. expm1-udef 64.3%

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

10. Applied egg-rr 64.3%

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

    1. expm1-def 87.5%

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

    2. expm1-log1p 90.9%

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

    3. +-commutative 90.9%

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

    4. +-commutative 90.9%

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

    5. distribute-lft-in 91.0%

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

    6. *-rgt-identity 91.0%

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

    7. rem-square-sqrt 91.0%

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

    8. hypot-def 99.6%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

    \[\leadsto \frac{{\left(x + 1\right)}^{-0.5}}{x + \mathsf{hypot}\left(x, \sqrt{x}\right)} \]
14. Add Preprocessing

Alternative 4: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.15)
   (+ (pow x -0.5) (/ -1.0 (+ 1.0 (* x 0.5))))
   (/ (/ 0.5 x) (sqrt (+ x 1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.15) {
		tmp = pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = (0.5 / x) / sqrt((x + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.15d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) / (1.0d0 + (x * 0.5d0)))
    else
        tmp = (0.5d0 / x) / sqrt((x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.15) {
		tmp = Math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = (0.5 / x) / Math.sqrt((x + 1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.15:
		tmp = math.pow(x, -0.5) + (-1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = (0.5 / x) / math.sqrt((x + 1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.15)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.15)
		tmp = (x ^ -0.5) + (-1.0 / (1.0 + (x * 0.5)));
	else
		tmp = (0.5 / x) / sqrt((x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.15], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1499999999999999

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 98.6%

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

      1. *-commutative 98.6%

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

   5. Simplified 98.6%

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

      1. expm1-log1p-u 91.7%

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

      2. expm1-udef 91.7%

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

      3. pow1/2 91.7%

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

      4. pow-flip 91.7%

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

      5. metadata-eval 91.7%

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

   7. Applied egg-rr 91.7%

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

      1. expm1-def 91.7%

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

      2. expm1-log1p 99.0%

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

   9. Simplified 99.0%

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

   if 1.1499999999999999 < x

   1. Initial program 36.5%

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

      1. frac-sub 36.6%

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

      2. associate-/r* 36.6%

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

      3. *-un-lft-identity 36.6%

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

      4. +-commutative 36.6%

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

      5. *-rgt-identity 36.6%

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

      6. +-commutative 36.6%

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

   4. Applied egg-rr 36.6%

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

      1. flip-- 36.3%

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

      2. div-inv 36.3%

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

      3. add-sqr-sqrt 20.5%

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

      4. add-sqr-sqrt 37.9%

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

      5. associate--l+ 98.7%

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

   6. Applied egg-rr 99.3%

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

      1. +-inverses 98.7%

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

      2. metadata-eval 98.7%

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

      3. *-lft-identity 98.7%

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

      4. +-commutative 98.7%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around inf 97.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.15:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.7)
   (+ (+ (pow x -0.5) (* x 0.5)) -1.0)
   (/ (/ 0.5 x) (sqrt (+ x 1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.7) {
		tmp = (pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (0.5 / x) / sqrt((x + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.7d0) then
        tmp = ((x ** (-0.5d0)) + (x * 0.5d0)) + (-1.0d0)
    else
        tmp = (0.5d0 / x) / sqrt((x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.7) {
		tmp = (Math.pow(x, -0.5) + (x * 0.5)) + -1.0;
	} else {
		tmp = (0.5 / x) / Math.sqrt((x + 1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.7:
		tmp = (math.pow(x, -0.5) + (x * 0.5)) + -1.0
	else:
		tmp = (0.5 / x) / math.sqrt((x + 1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.7)
		tmp = Float64(Float64((x ^ -0.5) + Float64(x * 0.5)) + -1.0);
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.7)
		tmp = ((x ^ -0.5) + (x * 0.5)) + -1.0;
	else
		tmp = (0.5 / x) / sqrt((x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.7], N[(N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.69999999999999996

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 98.9%

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

   if 0.69999999999999996 < x

   1. Initial program 36.5%

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

      1. frac-sub 36.6%

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

      2. associate-/r* 36.6%

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

      3. *-un-lft-identity 36.6%

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

      4. +-commutative 36.6%

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

      5. *-rgt-identity 36.6%

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

      6. +-commutative 36.6%

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

   4. Applied egg-rr 36.6%

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

      1. flip-- 36.3%

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

      2. div-inv 36.3%

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

      3. add-sqr-sqrt 20.5%

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

      4. add-sqr-sqrt 37.9%

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

      5. associate--l+ 98.7%

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

   6. Applied egg-rr 99.3%

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

      1. +-inverses 98.7%

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

      2. metadata-eval 98.7%

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

      3. *-lft-identity 98.7%

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

      4. +-commutative 98.7%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around inf 97.7%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.7:\\ \;\;\;\;\left({x}^{-0.5} + x \cdot 0.5\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot \left(x + 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.68) (+ (pow x -0.5) -1.0) (/ 1.0 (sqrt (* x (+ x 1.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / sqrt((x * (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = 1.0d0 / sqrt((x * (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = 1.0 / Math.sqrt((x * (x + 1.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.68:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = 1.0 / math.sqrt((x * (x + 1.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(1.0 / sqrt(Float64(x * Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = 1.0 / sqrt((x * (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.68], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 / N[Sqrt[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 98.3%

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

   if 0.680000000000000049 < x

   1. Initial program 36.5%

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

      1. frac-sub 36.6%

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

      2. *-un-lft-identity 36.6%

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

      3. +-commutative 36.6%

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

      4. *-rgt-identity 36.6%

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

      5. sqrt-unprod 36.6%

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

      6. +-commutative 36.6%

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

   4. Applied egg-rr 36.6%

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

   5. Taylor expanded in x around 0 35.5%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot \left(x + 1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.48) (+ (pow x -0.5) -1.0) (/ (/ 0.5 x) (sqrt (+ x 1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.48) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) / sqrt((x + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.48d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (0.5d0 / x) / sqrt((x + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.48) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = (0.5 / x) / Math.sqrt((x + 1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.48:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = (0.5 / x) / math.sqrt((x + 1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.48)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64(Float64(0.5 / x) / sqrt(Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.48)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (0.5 / x) / sqrt((x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.48], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(0.5 / x), $MachinePrecision] / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.47999999999999998

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 98.3%

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

   if 0.47999999999999998 < x

   1. Initial program 36.5%

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

      1. frac-sub 36.6%

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

      2. associate-/r* 36.6%

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

      3. *-un-lft-identity 36.6%

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

      4. +-commutative 36.6%

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

      5. *-rgt-identity 36.6%

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

      6. +-commutative 36.6%

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

   4. Applied egg-rr 36.6%

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

      1. flip-- 36.3%

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

      2. div-inv 36.3%

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

      3. add-sqr-sqrt 20.5%

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

      4. add-sqr-sqrt 37.9%

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

      5. associate--l+ 98.7%

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

   6. Applied egg-rr 99.3%

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

      1. +-inverses 98.7%

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

      2. metadata-eval 98.7%

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

      3. *-lft-identity 98.7%

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

      4. +-commutative 98.7%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in x around inf 97.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.48:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5}{x}}{\sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.599999999999999978

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

      8. mul0-lft 100.0%

         \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 98.3%

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

   if 0.599999999999999978 < x

   1. Initial program 36.5%

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

      1. frac-sub 36.6%

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

      2. *-un-lft-identity 36.6%

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

      3. +-commutative 36.6%

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

      4. *-rgt-identity 36.6%

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

      5. sqrt-unprod 36.6%

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

      6. +-commutative 36.6%

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

   4. Applied egg-rr 36.6%

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

   5. Taylor expanded in x around inf 35.8%

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

   6. Taylor expanded in x around 0 7.8%

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

4. Final simplification 53.4%

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

Alternative 9: 52.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

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

   1. *-un-lft-identity 68.3%

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

   2. clear-num 68.3%

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

   3. associate-/r/ 68.3%

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

   4. prod-diff 68.3%

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

   5. *-un-lft-identity 68.3%

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

   6. fma-neg 68.3%

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

   7. *-un-lft-identity 68.3%

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

   8. pow1/2 68.3%

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

   9. pow-flip 62.5%

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

   10. metadata-eval 62.5%

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

   11. pow1/2 62.5%

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

   12. pow-flip 68.5%

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

   13. +-commutative 68.5%

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

   14. metadata-eval 68.5%

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

4. Applied egg-rr 68.5%

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

   1. associate-+l- 68.5%

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

   2. expm1-log1p 68.5%

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

   3. expm1-def 53.9%

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

   4. associate--l- 53.9%

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

   5. fma-udef 53.9%

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

   6. distribute-lft1-in 53.9%

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

   7. metadata-eval 53.9%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

   8. mul0-lft 53.9%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

   9. metadata-eval 53.9%

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

   10. expm1-def 68.5%

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

   11. expm1-log1p 68.5%

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

6. Simplified 68.5%

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

8. Applied egg-rr 51.3%

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

   1. associate-*r/ 51.3%

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

   2. *-rgt-identity 51.3%

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

10. Simplified 51.3%

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

11. Taylor expanded in x around 0 52.4%

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

    1. distribute-rgt-in 52.4%

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

    2. *-lft-identity 52.4%

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

    3. pow-plus 52.6%

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

    4. metadata-eval 52.6%

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

13. Simplified 52.6%

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

    1. expm1-log1p-u 49.2%

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

    2. expm1-udef 62.3%

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

    3. pow1/2 62.3%

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

15. Applied egg-rr 62.3%

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

    1. expm1-def 49.2%

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

    2. expm1-log1p 52.6%

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

17. Simplified 52.6%

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

18. Final simplification 52.6%

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

Alternative 10: 50.4% 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 68.3%

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

   1. *-un-lft-identity 68.3%

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

   2. clear-num 68.3%

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

   3. associate-/r/ 68.3%

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

   4. prod-diff 68.3%

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

   5. *-un-lft-identity 68.3%

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

   6. fma-neg 68.3%

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

   7. *-un-lft-identity 68.3%

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

   8. pow1/2 68.3%

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

   9. pow-flip 62.5%

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

   10. metadata-eval 62.5%

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

   11. pow1/2 62.5%

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

   12. pow-flip 68.5%

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

   13. +-commutative 68.5%

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

   14. metadata-eval 68.5%

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

4. Applied egg-rr 68.5%

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

   1. associate-+l- 68.5%

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

   2. expm1-log1p 68.5%

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

   3. expm1-def 53.9%

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

   4. associate--l- 53.9%

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

   5. fma-udef 53.9%

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

   6. distribute-lft1-in 53.9%

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

   7. metadata-eval 53.9%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

   8. mul0-lft 53.9%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

   9. metadata-eval 53.9%

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

   10. expm1-def 68.5%

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

   11. expm1-log1p 68.5%

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

6. Simplified 68.5%

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

8. Applied egg-rr 51.3%

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

   1. associate-*r/ 51.3%

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

   2. *-rgt-identity 51.3%

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

10. Simplified 51.3%

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

11. Taylor expanded in x around inf 50.0%

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

    1. unpow1/2 50.0%

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

    2. rem-exp-log 46.7%

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

    3. exp-neg 46.7%

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

    4. exp-prod 46.7%

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

    5. distribute-lft-neg-out 46.7%

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

    6. distribute-rgt-neg-in 46.7%

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

    7. metadata-eval 46.7%

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

    8. exp-to-pow 50.1%

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

13. Simplified 50.1%

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

14. Final simplification 50.1%

    \[\leadsto {x}^{-0.5} \]
15. Add Preprocessing

Alternative 11: 7.4% accurate, 41.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + 0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ x 0.5)))

C

double code(double x) {
	return 1.0 / (x + 0.5);
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (x + 0.5);
}

Python

def code(x):
	return 1.0 / (x + 0.5)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

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

   1. frac-sub 68.3%

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

   2. *-un-lft-identity 68.3%

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

   3. +-commutative 68.3%

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

   4. *-rgt-identity 68.3%

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

   5. sqrt-unprod 68.3%

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

   6. +-commutative 68.3%

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

4. Applied egg-rr 68.3%

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

5. Taylor expanded in x around inf 21.4%

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

6. Taylor expanded in x around 0 7.5%

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

7. Final simplification 7.5%

   \[\leadsto \frac{1}{x + 0.5} \]
8. Add Preprocessing

Alternative 12: 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 68.3%

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

   1. *-un-lft-identity 68.3%

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

   2. clear-num 68.3%

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

   3. associate-/r/ 68.3%

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

   4. prod-diff 68.3%

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

   5. *-un-lft-identity 68.3%

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

   6. fma-neg 68.3%

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

   7. *-un-lft-identity 68.3%

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

   8. pow1/2 68.3%

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

   9. pow-flip 62.5%

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

   10. metadata-eval 62.5%

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

   11. pow1/2 62.5%

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

   12. pow-flip 68.5%

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

   13. +-commutative 68.5%

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

   14. metadata-eval 68.5%

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

4. Applied egg-rr 68.5%

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

   1. associate-+l- 68.5%

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

   2. expm1-log1p 68.5%

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

   3. expm1-def 53.9%

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

   4. associate--l- 53.9%

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

   5. fma-udef 53.9%

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

   6. distribute-lft1-in 53.9%

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

   7. metadata-eval 53.9%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0} \cdot {\left(1 + x\right)}^{-0.5}\right)\right) \]

   8. mul0-lft 53.9%

      \[\leadsto {x}^{-0.5} - \left(e^{\mathsf{log1p}\left({\left(1 + x\right)}^{-0.5}\right)} - \left(1 + \color{blue}{0}\right)\right) \]

   9. metadata-eval 53.9%

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

   10. expm1-def 68.5%

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

   11. expm1-log1p 68.5%

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

6. Simplified 68.5%

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

8. Applied egg-rr 51.3%

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

   1. associate-*r/ 51.3%

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

   2. *-rgt-identity 51.3%

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

10. Simplified 51.3%

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

11. Taylor expanded in x around 0 52.4%

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

    1. distribute-rgt-in 52.4%

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

    2. *-lft-identity 52.4%

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

    3. pow-plus 52.6%

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

    4. metadata-eval 52.6%

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

13. Simplified 52.6%

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

14. Taylor expanded in x around inf 7.5%

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

15. Final simplification 7.5%

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

Alternative 13: 1.9% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 68.3%

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

3. Taylor expanded in x around 0 50.6%

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

4. Taylor expanded in x around inf 1.9%

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

5. Final simplification 1.9%

   \[\leadsto -1 \]
6. Add Preprocessing

Alternative 14: 5.7% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 68.3%

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

   1. frac-sub 68.3%

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

   2. *-un-lft-identity 68.3%

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

   3. +-commutative 68.3%

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

   4. *-rgt-identity 68.3%

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

   5. sqrt-unprod 68.3%

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

   6. +-commutative 68.3%

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

4. Applied egg-rr 68.3%

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

5. Taylor expanded in x around inf 21.4%

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

6. Taylor expanded in x around 0 5.9%

   \[\leadsto \color{blue}{2} \]

7. Final simplification 5.9%

   \[\leadsto 2 \]
8. Add Preprocessing

Developer target: 98.9% 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 2024021 
(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)))))