2isqrt (example 3.6)

Percentage Accurate: 68.8% → 99.8%

Time: 13.7s

Alternatives: 13

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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 0:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\left(x + 1\right) + \sqrt{x \cdot \left(x + 1\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 0.0)
   (* (pow x -0.5) (/ 0.5 x))
   (*
    (pow x -0.5)
    (/ (+ 1.0 (* x 0.0)) (+ (+ x 1.0) (sqrt (* x (+ x 1.0))))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 0.0) {
		tmp = pow(x, -0.5) * (0.5 / x);
	} else {
		tmp = pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x + 1.0) + sqrt((x * (x + 1.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((x + 1.0d0)))) <= 0.0d0) then
        tmp = (x ** (-0.5d0)) * (0.5d0 / x)
    else
        tmp = (x ** (-0.5d0)) * ((1.0d0 + (x * 0.0d0)) / ((x + 1.0d0) + sqrt((x * (x + 1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((x + 1.0)))) <= 0.0) {
		tmp = Math.pow(x, -0.5) * (0.5 / x);
	} else {
		tmp = Math.pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x + 1.0) + Math.sqrt((x * (x + 1.0)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 0.0:
		tmp = math.pow(x, -0.5) * (0.5 / x)
	else:
		tmp = math.pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x + 1.0) + math.sqrt((x * (x + 1.0)))))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 / x), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(1.0 + N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[Sqrt[N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 34.3%

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

      1. frac-sub 34.3%

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

      2. *-un-lft-identity 34.3%

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

      3. +-commutative 34.3%

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

      4. *-rgt-identity 34.3%

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

      5. sqrt-unprod 34.3%

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

      6. +-commutative 34.3%

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

   4. Applied egg-rr 34.3%

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

      1. flip-- 34.3%

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

      2. add-sqr-sqrt 34.4%

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

      3. +-commutative 34.4%

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

      4. add-sqr-sqrt 34.3%

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

      5. +-commutative 34.3%

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

   6. Applied egg-rr 34.3%

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

      1. *-un-lft-identity 34.3%

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

      2. sqrt-prod 34.3%

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

      3. +-commutative 34.3%

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

      4. times-frac 34.3%

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

      5. pow1/2 34.3%

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

      6. pow-flip 34.3%

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

      7. metadata-eval 34.3%

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

      8. associate--l+ 34.3%

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

   8. Applied egg-rr 34.3%

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

      1. associate-/l/ 34.3%

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

      2. +-commutative 34.3%

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

      3. sub-neg 34.3%

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

      4. associate-+l+ 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. distribute-lft-out 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-rgt-in 99.3%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \sqrt{x} \cdot \sqrt{x + 1}}} \]

      11. rem-square-sqrt 99.4%

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

      12. +-commutative 99.4%

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

      13. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 99.6%

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

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

   1. Initial program 97.8%

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

      1. frac-sub 97.9%

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

      2. *-un-lft-identity 97.9%

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

      3. +-commutative 97.9%

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

      4. *-rgt-identity 97.9%

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

      5. sqrt-unprod 97.9%

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

      6. +-commutative 97.9%

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

   4. Applied egg-rr 97.9%

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

      1. flip-- 97.8%

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

      2. add-sqr-sqrt 98.6%

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

      3. +-commutative 98.6%

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

      4. add-sqr-sqrt 99.4%

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

      5. +-commutative 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. sqrt-prod 99.4%

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

      3. +-commutative 99.4%

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

      4. times-frac 99.4%

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

      5. pow1/2 99.4%

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

      6. pow-flip 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate--l+ 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-/l/ 99.9%

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

      2. +-commutative 99.9%

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

      3. sub-neg 99.9%

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

      4. associate-+l+ 99.9%

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

      5. neg-mul-1 99.9%

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

      6. *-commutative 99.9%

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

      7. *-rgt-identity 99.9%

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

      8. distribute-lft-out 99.9%

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

      9. metadata-eval 99.9%

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

      10. distribute-rgt-in 99.9%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \sqrt{x} \cdot \sqrt{x + 1}}} \]

      11. rem-square-sqrt 99.9%

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

      12. +-commutative 99.9%

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

      13. +-commutative 99.9%

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

   10. Simplified 99.9%

       \[\leadsto \color{blue}{{x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\left(1 + x\right) + \sqrt{x} \cdot \sqrt{1 + x}}} \]
   11. Step-by-step derivation

       1. expm1-log1p-u 99.9%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\left(1 + x\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x} \cdot \sqrt{1 + x}\right)\right)}} \]

       2. expm1-udef 99.9%

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

       3. sqrt-unprod 99.9%

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

       4. +-commutative 99.9%

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

   12. Applied egg-rr 99.9%

       \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\left(1 + x\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x \cdot \left(x + 1\right)}\right)} - 1\right)}} \]
   13. Step-by-step derivation

       1. expm1-def 99.9%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\left(1 + x\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x \cdot \left(x + 1\right)}\right)\right)}} \]

       2. expm1-log1p 99.9%

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

   14. Simplified 99.9%

       \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\left(1 + x\right) + \color{blue}{\sqrt{x \cdot \left(x + 1\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 10^{-13}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{1 + x \cdot 0}{x \cdot 2 + 1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{x + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ x 1.0)))) 1e-13)
   (* (pow x -0.5) (/ (+ 1.0 (* x 0.0)) (+ (* x 2.0) 1.5)))
   (- (pow x -0.5) (sqrt (/ 1.0 (+ x 1.0))))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 1e-13) {
		tmp = pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x * 2.0) + 1.5));
	} else {
		tmp = pow(x, -0.5) - sqrt((1.0 / (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((x + 1.0d0)))) <= 1d-13) then
        tmp = (x ** (-0.5d0)) * ((1.0d0 + (x * 0.0d0)) / ((x * 2.0d0) + 1.5d0))
    else
        tmp = (x ** (-0.5d0)) - sqrt((1.0d0 / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((x + 1.0)))) <= 1e-13) {
		tmp = Math.pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x * 2.0) + 1.5));
	} else {
		tmp = Math.pow(x, -0.5) - Math.sqrt((1.0 / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((x + 1.0)))) <= 1e-13:
		tmp = math.pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x * 2.0) + 1.5))
	else:
		tmp = math.pow(x, -0.5) - math.sqrt((1.0 / (x + 1.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(x + 1.0)))) <= 1e-13)
		tmp = Float64((x ^ -0.5) * Float64(Float64(1.0 + Float64(x * 0.0)) / Float64(Float64(x * 2.0) + 1.5)));
	else
		tmp = Float64((x ^ -0.5) - sqrt(Float64(1.0 / Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((x + 1.0)))) <= 1e-13)
		tmp = (x ^ -0.5) * ((1.0 + (x * 0.0)) / ((x * 2.0) + 1.5));
	else
		tmp = (x ^ -0.5) - sqrt((1.0 / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-13], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(1.0 + N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Sqrt[N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 1e-13

   1. Initial program 34.5%

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

      1. frac-sub 34.6%

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

      2. *-un-lft-identity 34.6%

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

      3. +-commutative 34.6%

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

      4. *-rgt-identity 34.6%

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

      5. sqrt-unprod 34.6%

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

      6. +-commutative 34.6%

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

   4. Applied egg-rr 34.6%

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

      1. flip-- 34.6%

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

      2. add-sqr-sqrt 35.3%

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

      3. +-commutative 35.3%

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

      4. add-sqr-sqrt 35.7%

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

      5. +-commutative 35.7%

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

   6. Applied egg-rr 35.7%

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

      1. *-un-lft-identity 35.7%

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

      2. sqrt-prod 35.7%

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

      3. +-commutative 35.7%

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

      4. times-frac 35.7%

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

      5. pow1/2 35.7%

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

      6. pow-flip 35.7%

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

      7. metadata-eval 35.7%

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

      8. associate--l+ 35.7%

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

   8. Applied egg-rr 35.7%

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

      1. associate-/l/ 35.7%

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

      2. +-commutative 35.7%

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

      3. sub-neg 35.7%

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

      4. associate-+l+ 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. distribute-lft-out 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-rgt-in 99.3%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \sqrt{x} \cdot \sqrt{x + 1}}} \]

      11. rem-square-sqrt 99.4%

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

      12. +-commutative 99.4%

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

      13. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 99.6%

       \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{1.5 + 2 \cdot x}} \]
   12. Step-by-step derivation

       1. +-commutative 99.6%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{2 \cdot x + 1.5}} \]

       2. *-commutative 99.6%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{x \cdot 2} + 1.5} \]

   13. Simplified 99.6%

       \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{x \cdot 2 + 1.5}} \]

   if 1e-13 < (-.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. add-sqr-sqrt 99.2%

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

      2. sqrt-unprod 99.2%

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

      3. frac-times 99.2%

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

      4. metadata-eval 99.2%

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

      5. add-sqr-sqrt 99.2%

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

      6. +-commutative 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. expm1-log1p-u 89.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      2. expm1-udef 89.4%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)} - 1\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      3. pow1/2 89.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{x}^{0.5}}}\right)} - 1\right) - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      4. pow-flip 89.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(-0.5\right)}}\right)} - 1\right) - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      5. metadata-eval 89.4%

         \[\leadsto \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

   6. Applied egg-rr 91.9%

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

      1. expm1-def 89.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      2. expm1-log1p 97.0%

         \[\leadsto \color{blue}{{x}^{-0.5}} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

   8. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{x + 1}} \leq 10^{-13}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{1 + x \cdot 0}{x \cdot 2 + 1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt{\frac{1}{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. frac-sub 63.6%

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

   2. *-un-lft-identity 63.6%

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

   3. +-commutative 63.6%

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

   4. *-rgt-identity 63.6%

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

   5. sqrt-unprod 63.6%

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

   6. +-commutative 63.6%

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

4. Applied egg-rr 63.6%

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

   1. flip-- 63.6%

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

   2. add-sqr-sqrt 64.0%

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

   3. +-commutative 64.0%

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

   4. add-sqr-sqrt 64.3%

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

   5. +-commutative 64.3%

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

6. Applied egg-rr 64.3%

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

   1. *-un-lft-identity 64.3%

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

   2. sqrt-prod 64.3%

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

   3. +-commutative 64.3%

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

   4. times-frac 64.3%

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

   5. pow1/2 64.3%

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

   6. pow-flip 64.5%

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

   7. metadata-eval 64.5%

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

   8. associate--l+ 64.5%

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

8. Applied egg-rr 64.5%

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

   1. associate-/l/ 64.5%

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

   2. +-commutative 64.5%

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

   3. sub-neg 64.5%

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

   4. associate-+l+ 99.6%

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

   5. neg-mul-1 99.6%

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

   6. *-commutative 99.6%

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

   7. *-rgt-identity 99.6%

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

   8. distribute-lft-out 99.6%

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

   9. metadata-eval 99.6%

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

   10. distribute-rgt-in 99.6%

       \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \sqrt{x} \cdot \sqrt{x + 1}}} \]

   11. rem-square-sqrt 99.7%

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

   12. +-commutative 99.7%

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

   13. +-commutative 99.7%

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

10. Simplified 99.7%

    \[\leadsto \color{blue}{{x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\left(1 + x\right) + \sqrt{x} \cdot \sqrt{1 + x}}} \]
11. Step-by-step derivation

    1. expm1-log1p-u 96.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\left(1 + x\right) + \sqrt{x} \cdot \sqrt{1 + x}}\right)\right)} \]

    2. expm1-udef 60.0%

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

    3. mul0-rgt 60.0%

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

    4. metadata-eval 60.0%

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

    5. +-commutative 60.0%

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

    6. +-commutative 60.0%

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

    7. fma-def 60.0%

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

    8. +-commutative 60.0%

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

12. Applied egg-rr 60.0%

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

    1. expm1-def 96.3%

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

    2. expm1-log1p 99.8%

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

    3. associate-*r/ 99.8%

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

    4. *-rgt-identity 99.8%

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

14. Simplified 99.8%

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

15. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (pow x -0.5)
  (/ (+ 1.0 (* x 0.0)) (+ (+ x 1.0) (* (sqrt x) (sqrt (+ x 1.0)))))))

C

double code(double x) {
	return pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x + 1.0) + (sqrt(x) * sqrt((x + 1.0)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** (-0.5d0)) * ((1.0d0 + (x * 0.0d0)) / ((x + 1.0d0) + (sqrt(x) * sqrt((x + 1.0d0)))))
end function

Java

public static double code(double x) {
	return Math.pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x + 1.0) + (Math.sqrt(x) * Math.sqrt((x + 1.0)))));
}

Python

def code(x):
	return math.pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x + 1.0) + (math.sqrt(x) * math.sqrt((x + 1.0)))))

Julia

function code(x)
	return Float64((x ^ -0.5) * Float64(Float64(1.0 + Float64(x * 0.0)) / Float64(Float64(x + 1.0) + Float64(sqrt(x) * sqrt(Float64(x + 1.0))))))
end

MATLAB

function tmp = code(x)
	tmp = (x ^ -0.5) * ((1.0 + (x * 0.0)) / ((x + 1.0) + (sqrt(x) * sqrt((x + 1.0)))));
end

Wolfram

code[x_] := N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(1.0 + N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.6%

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

   1. frac-sub 63.6%

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

   2. *-un-lft-identity 63.6%

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

   3. +-commutative 63.6%

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

   4. *-rgt-identity 63.6%

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

   5. sqrt-unprod 63.6%

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

   6. +-commutative 63.6%

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

4. Applied egg-rr 63.6%

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

   1. flip-- 63.6%

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

   2. add-sqr-sqrt 64.0%

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

   3. +-commutative 64.0%

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

   4. add-sqr-sqrt 64.3%

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

   5. +-commutative 64.3%

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

6. Applied egg-rr 64.3%

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

   1. *-un-lft-identity 64.3%

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

   2. sqrt-prod 64.3%

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

   3. +-commutative 64.3%

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

   4. times-frac 64.3%

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

   5. pow1/2 64.3%

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

   6. pow-flip 64.5%

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

   7. metadata-eval 64.5%

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

   8. associate--l+ 64.5%

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

8. Applied egg-rr 64.5%

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

   1. associate-/l/ 64.5%

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

   2. +-commutative 64.5%

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

   3. sub-neg 64.5%

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

   4. associate-+l+ 99.6%

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

   5. neg-mul-1 99.6%

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

   6. *-commutative 99.6%

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

   7. *-rgt-identity 99.6%

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

   8. distribute-lft-out 99.6%

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

   9. metadata-eval 99.6%

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

   10. distribute-rgt-in 99.6%

       \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \sqrt{x} \cdot \sqrt{x + 1}}} \]

   11. rem-square-sqrt 99.7%

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

   12. +-commutative 99.7%

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

   13. +-commutative 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

    \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\left(x + 1\right) + \sqrt{x} \cdot \sqrt{x + 1}} \]
12. Add Preprocessing

Alternative 5: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 84000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{1 + x \cdot 0}{x \cdot 2 + 1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 84000.0)
   (- (pow x -0.5) (pow (+ x 1.0) -0.5))
   (* (pow x -0.5) (/ (+ 1.0 (* x 0.0)) (+ (* x 2.0) 1.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 84000.0) {
		tmp = pow(x, -0.5) - pow((x + 1.0), -0.5);
	} else {
		tmp = pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x * 2.0) + 1.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 84000.0d0) then
        tmp = (x ** (-0.5d0)) - ((x + 1.0d0) ** (-0.5d0))
    else
        tmp = (x ** (-0.5d0)) * ((1.0d0 + (x * 0.0d0)) / ((x * 2.0d0) + 1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 84000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((x + 1.0), -0.5);
	} else {
		tmp = Math.pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x * 2.0) + 1.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 84000.0:
		tmp = math.pow(x, -0.5) - math.pow((x + 1.0), -0.5)
	else:
		tmp = math.pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x * 2.0) + 1.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 84000.0)
		tmp = Float64((x ^ -0.5) - (Float64(x + 1.0) ^ -0.5));
	else
		tmp = Float64((x ^ -0.5) * Float64(Float64(1.0 + Float64(x * 0.0)) / Float64(Float64(x * 2.0) + 1.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 84000.0)
		tmp = (x ^ -0.5) - ((x + 1.0) ^ -0.5);
	else
		tmp = (x ^ -0.5) * ((1.0 + (x * 0.0)) / ((x * 2.0) + 1.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 84000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(x + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(1.0 + N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 84000

   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 99.7%

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

      10. metadata-eval 99.7%

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

      11. pow1/2 99.7%

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

      12. pow-flip 99.7%

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

      13. +-commutative 99.7%

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

      14. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

      2. sub-neg 99.7%

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

      3. fma-udef 99.7%

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

      4. distribute-lft1-in 99.7%

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

      5. metadata-eval 99.7%

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

      6. mul0-lft 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+r+ 99.7%

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

      9. sub-neg 99.7%

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

      10. neg-sub0 99.7%

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

      11. +-commutative 99.7%

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

      12. sub-neg 99.7%

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

   6. Simplified 99.7%

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

   if 84000 < x

   1. Initial program 34.5%

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

      1. frac-sub 34.6%

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

      2. *-un-lft-identity 34.6%

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

      3. +-commutative 34.6%

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

      4. *-rgt-identity 34.6%

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

      5. sqrt-unprod 34.6%

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

      6. +-commutative 34.6%

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

   4. Applied egg-rr 34.6%

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

      1. flip-- 34.6%

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

      2. add-sqr-sqrt 35.3%

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

      3. +-commutative 35.3%

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

      4. add-sqr-sqrt 35.7%

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

      5. +-commutative 35.7%

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

   6. Applied egg-rr 35.7%

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

      1. *-un-lft-identity 35.7%

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

      2. sqrt-prod 35.7%

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

      3. +-commutative 35.7%

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

      4. times-frac 35.7%

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

      5. pow1/2 35.7%

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

      6. pow-flip 35.7%

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

      7. metadata-eval 35.7%

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

      8. associate--l+ 35.7%

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

   8. Applied egg-rr 35.7%

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

      1. associate-/l/ 35.7%

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

      2. +-commutative 35.7%

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

      3. sub-neg 35.7%

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

      4. associate-+l+ 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. distribute-lft-out 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-rgt-in 99.3%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \sqrt{x} \cdot \sqrt{x + 1}}} \]

      11. rem-square-sqrt 99.4%

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

      12. +-commutative 99.4%

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

      13. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 99.6%

       \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{1.5 + 2 \cdot x}} \]
   12. Step-by-step derivation

       1. +-commutative 99.6%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{2 \cdot x + 1.5}} \]

       2. *-commutative 99.6%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{x \cdot 2} + 1.5} \]

   13. Simplified 99.6%

       \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{x \cdot 2 + 1.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 84000:\\ \;\;\;\;{x}^{-0.5} - {\left(x + 1\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{1 + x \cdot 0}{x \cdot 2 + 1.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{1 + x \cdot 0}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (+ (pow x -0.5) (- -1.0 (* x (+ -0.5 (* x 0.375)))))
   (* (pow x -0.5) (/ (+ 1.0 (* x 0.0)) (+ 1.5 (- (* x 2.0) (/ 0.125 x)))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = pow(x, -0.5) * ((1.0 + (x * 0.0)) / (1.5 + ((x * 2.0) - (0.125 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) - (x * ((-0.5d0) + (x * 0.375d0))))
    else
        tmp = (x ** (-0.5d0)) * ((1.0d0 + (x * 0.0d0)) / (1.5d0 + ((x * 2.0d0) - (0.125d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = Math.pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = Math.pow(x, -0.5) * ((1.0 + (x * 0.0)) / (1.5 + ((x * 2.0) - (0.125 / x))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = math.pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))))
	else:
		tmp = math.pow(x, -0.5) * ((1.0 + (x * 0.0)) / (1.5 + ((x * 2.0) - (0.125 / x))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 - Float64(x * Float64(-0.5 + Float64(x * 0.375)))));
	else
		tmp = Float64((x ^ -0.5) * Float64(Float64(1.0 + Float64(x * 0.0)) / Float64(1.5 + Float64(Float64(x * 2.0) - Float64(0.125 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.5)
		tmp = (x ^ -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	else
		tmp = (x ^ -0.5) * ((1.0 + (x * 0.0)) / (1.5 + ((x * 2.0) - (0.125 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 - N[(x * N[(-0.5 + N[(x * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(1.0 + N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / N[(1.5 + N[(N[(x * 2.0), $MachinePrecision] - N[(0.125 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.1%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(1 + \left(-0.5 \cdot x + 0.375 \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(\color{blue}{x \cdot -0.5} + 0.375 \cdot {x}^{2}\right)\right) \]

      2. *-commutative 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{{x}^{2} \cdot 0.375}\right)\right) \]

      3. unpow2 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{\left(x \cdot x\right)} \cdot 0.375\right)\right) \]

      4. associate-*l* 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{x \cdot \left(x \cdot 0.375\right)}\right)\right) \]

      5. distribute-lft-out 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \color{blue}{x \cdot \left(-0.5 + x \cdot 0.375\right)}\right) \]

   5. Simplified 99.1%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right)} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 91.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      2. expm1-udef 91.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)} - 1\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      3. pow1/2 91.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{x}^{0.5}}}\right)} - 1\right) - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      4. pow-flip 91.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(-0.5\right)}}\right)} - 1\right) - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      5. metadata-eval 91.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

   7. Applied egg-rr 91.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]
   8. Step-by-step derivation

      1. expm1-def 91.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      2. expm1-log1p 99.6%

         \[\leadsto \color{blue}{{x}^{-0.5}} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

   9. Simplified 99.6%

      \[\leadsto \color{blue}{{x}^{-0.5}} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

   if 0.5 < x

   1. Initial program 35.6%

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

      1. flip-- 35.7%

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

      2. add-sqr-sqrt 36.4%

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

      3. +-commutative 36.4%

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

      4. add-sqr-sqrt 37.0%

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

      5. +-commutative 37.0%

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

   6. Applied egg-rr 37.0%

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

      1. *-un-lft-identity 37.0%

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

      2. sqrt-prod 37.0%

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

      3. +-commutative 37.0%

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

      4. times-frac 37.0%

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

      5. pow1/2 37.0%

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

      6. pow-flip 37.0%

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

      7. metadata-eval 37.0%

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

      8. associate--l+ 37.0%

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

   8. Applied egg-rr 37.0%

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

      1. associate-/l/ 37.0%

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

      2. +-commutative 37.0%

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

      3. sub-neg 37.0%

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

      4. associate-+l+ 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. distribute-lft-out 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-rgt-in 99.3%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \sqrt{x} \cdot \sqrt{x + 1}}} \]

      11. rem-square-sqrt 99.4%

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

      12. +-commutative 99.4%

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

      13. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 98.8%

       \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{\left(1.5 + 2 \cdot x\right) - 0.125 \cdot \frac{1}{x}}} \]
   12. Step-by-step derivation

       1. associate--l+ 98.8%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{1.5 + \left(2 \cdot x - 0.125 \cdot \frac{1}{x}\right)}} \]

       2. *-commutative 98.8%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{1.5 + \left(\color{blue}{x \cdot 2} - 0.125 \cdot \frac{1}{x}\right)} \]

       3. associate-*r/ 98.8%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{1.5 + \left(x \cdot 2 - \color{blue}{\frac{0.125 \cdot 1}{x}}\right)} \]

       4. metadata-eval 98.8%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{1.5 + \left(x \cdot 2 - \frac{\color{blue}{0.125}}{x}\right)} \]

   13. Simplified 98.8%

       \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{1 + x \cdot 0}{1.5 + \left(x \cdot 2 - \frac{0.125}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{1 + x \cdot 0}{x \cdot 2 + 1.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (+ (pow x -0.5) (- -1.0 (* x (+ -0.5 (* x 0.375)))))
   (* (pow x -0.5) (/ (+ 1.0 (* x 0.0)) (+ (* x 2.0) 1.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x * 2.0) + 1.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) - (x * ((-0.5d0) + (x * 0.375d0))))
    else
        tmp = (x ** (-0.5d0)) * ((1.0d0 + (x * 0.0d0)) / ((x * 2.0d0) + 1.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = Math.pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = Math.pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x * 2.0) + 1.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = math.pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))))
	else:
		tmp = math.pow(x, -0.5) * ((1.0 + (x * 0.0)) / ((x * 2.0) + 1.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 - Float64(x * Float64(-0.5 + Float64(x * 0.375)))));
	else
		tmp = Float64((x ^ -0.5) * Float64(Float64(1.0 + Float64(x * 0.0)) / Float64(Float64(x * 2.0) + 1.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.5)
		tmp = (x ^ -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	else
		tmp = (x ^ -0.5) * ((1.0 + (x * 0.0)) / ((x * 2.0) + 1.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 - N[(x * N[(-0.5 + N[(x * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(1.0 + N[(x * 0.0), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.1%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(1 + \left(-0.5 \cdot x + 0.375 \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(\color{blue}{x \cdot -0.5} + 0.375 \cdot {x}^{2}\right)\right) \]

      2. *-commutative 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{{x}^{2} \cdot 0.375}\right)\right) \]

      3. unpow2 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{\left(x \cdot x\right)} \cdot 0.375\right)\right) \]

      4. associate-*l* 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{x \cdot \left(x \cdot 0.375\right)}\right)\right) \]

      5. distribute-lft-out 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \color{blue}{x \cdot \left(-0.5 + x \cdot 0.375\right)}\right) \]

   5. Simplified 99.1%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right)} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 91.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      2. expm1-udef 91.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)} - 1\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      3. pow1/2 91.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{x}^{0.5}}}\right)} - 1\right) - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      4. pow-flip 91.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(-0.5\right)}}\right)} - 1\right) - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      5. metadata-eval 91.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

   7. Applied egg-rr 91.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]
   8. Step-by-step derivation

      1. expm1-def 91.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      2. expm1-log1p 99.6%

         \[\leadsto \color{blue}{{x}^{-0.5}} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

   9. Simplified 99.6%

      \[\leadsto \color{blue}{{x}^{-0.5}} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

   if 0.5 < x

   1. Initial program 35.6%

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

      1. flip-- 35.7%

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

      2. add-sqr-sqrt 36.4%

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

      3. +-commutative 36.4%

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

      4. add-sqr-sqrt 37.0%

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

      5. +-commutative 37.0%

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

   6. Applied egg-rr 37.0%

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

      1. *-un-lft-identity 37.0%

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

      2. sqrt-prod 37.0%

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

      3. +-commutative 37.0%

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

      4. times-frac 37.0%

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

      5. pow1/2 37.0%

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

      6. pow-flip 37.0%

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

      7. metadata-eval 37.0%

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

      8. associate--l+ 37.0%

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

   8. Applied egg-rr 37.0%

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

      1. associate-/l/ 37.0%

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

      2. +-commutative 37.0%

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

      3. sub-neg 37.0%

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

      4. associate-+l+ 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. distribute-lft-out 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-rgt-in 99.3%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \sqrt{x} \cdot \sqrt{x + 1}}} \]

      11. rem-square-sqrt 99.4%

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

      12. +-commutative 99.4%

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

      13. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 98.6%

       \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{1.5 + 2 \cdot x}} \]
   12. Step-by-step derivation

       1. +-commutative 98.6%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{2 \cdot x + 1.5}} \]

       2. *-commutative 98.6%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{x \cdot 2} + 1.5} \]

   13. Simplified 98.6%

       \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{x \cdot 2 + 1.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{1 + x \cdot 0}{x \cdot 2 + 1.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.95)
   (+ (pow x -0.5) (- -1.0 (* x (+ -0.5 (* x 0.375)))))
   (* (pow x -0.5) (/ 0.5 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.95) {
		tmp = pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.95d0) then
        tmp = (x ** (-0.5d0)) + ((-1.0d0) - (x * ((-0.5d0) + (x * 0.375d0))))
    else
        tmp = (x ** (-0.5d0)) * (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.95) {
		tmp = Math.pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	} else {
		tmp = Math.pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.95:
		tmp = math.pow(x, -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))))
	else:
		tmp = math.pow(x, -0.5) * (0.5 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.95)
		tmp = Float64((x ^ -0.5) + Float64(-1.0 - Float64(x * Float64(-0.5 + Float64(x * 0.375)))));
	else
		tmp = Float64((x ^ -0.5) * Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.95)
		tmp = (x ^ -0.5) + (-1.0 - (x * (-0.5 + (x * 0.375))));
	else
		tmp = (x ^ -0.5) * (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.95], N[(N[Power[x, -0.5], $MachinePrecision] + N[(-1.0 - N[(x * N[(-0.5 + N[(x * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.94999999999999996

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0 99.1%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(1 + \left(-0.5 \cdot x + 0.375 \cdot {x}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(\color{blue}{x \cdot -0.5} + 0.375 \cdot {x}^{2}\right)\right) \]

      2. *-commutative 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{{x}^{2} \cdot 0.375}\right)\right) \]

      3. unpow2 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{\left(x \cdot x\right)} \cdot 0.375\right)\right) \]

      4. associate-*l* 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \left(x \cdot -0.5 + \color{blue}{x \cdot \left(x \cdot 0.375\right)}\right)\right) \]

      5. distribute-lft-out 99.1%

         \[\leadsto \frac{1}{\sqrt{x}} - \left(1 + \color{blue}{x \cdot \left(-0.5 + x \cdot 0.375\right)}\right) \]

   5. Simplified 99.1%

      \[\leadsto \frac{1}{\sqrt{x}} - \color{blue}{\left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right)} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 91.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      2. expm1-udef 91.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{x}}\right)} - 1\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      3. pow1/2 91.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{x}^{0.5}}}\right)} - 1\right) - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      4. pow-flip 91.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(-0.5\right)}}\right)} - 1\right) - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      5. metadata-eval 91.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

   7. Applied egg-rr 91.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]
   8. Step-by-step derivation

      1. expm1-def 91.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

      2. expm1-log1p 99.6%

         \[\leadsto \color{blue}{{x}^{-0.5}} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

   9. Simplified 99.6%

      \[\leadsto \color{blue}{{x}^{-0.5}} - \left(1 + x \cdot \left(-0.5 + x \cdot 0.375\right)\right) \]

   if 0.94999999999999996 < x

   1. Initial program 35.6%

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

      1. flip-- 35.7%

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

      2. add-sqr-sqrt 36.4%

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

      3. +-commutative 36.4%

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

      4. add-sqr-sqrt 37.0%

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

      5. +-commutative 37.0%

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

   6. Applied egg-rr 37.0%

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

      1. *-un-lft-identity 37.0%

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

      2. sqrt-prod 37.0%

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

      3. +-commutative 37.0%

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

      4. times-frac 37.0%

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

      5. pow1/2 37.0%

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

      6. pow-flip 37.0%

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

      7. metadata-eval 37.0%

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

      8. associate--l+ 37.0%

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

   8. Applied egg-rr 37.0%

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

      1. associate-/l/ 37.0%

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

      2. +-commutative 37.0%

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

      3. sub-neg 37.0%

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

      4. associate-+l+ 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. distribute-lft-out 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-rgt-in 99.3%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \sqrt{x} \cdot \sqrt{x + 1}}} \]

      11. rem-square-sqrt 99.4%

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

      12. +-commutative 99.4%

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

      13. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 97.6%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.95:\\ \;\;\;\;{x}^{-0.5} + \left(-1 - x \cdot \left(-0.5 + x \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if 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 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. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul0-lft 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+r+ 100.0%

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

      9. sub-neg 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 99.5%

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

   if 1 < x

   1. Initial program 35.6%

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

      1. flip-- 35.7%

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

      2. add-sqr-sqrt 36.4%

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

      3. +-commutative 36.4%

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

      4. add-sqr-sqrt 37.0%

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

      5. +-commutative 37.0%

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

   6. Applied egg-rr 37.0%

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

      1. *-un-lft-identity 37.0%

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

      2. sqrt-prod 37.0%

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

      3. +-commutative 37.0%

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

      4. times-frac 37.0%

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

      5. pow1/2 37.0%

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

      6. pow-flip 37.0%

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

      7. metadata-eval 37.0%

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

      8. associate--l+ 37.0%

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

   8. Applied egg-rr 37.0%

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

      1. associate-/l/ 37.0%

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

      2. +-commutative 37.0%

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

      3. sub-neg 37.0%

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

      4. associate-+l+ 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. distribute-lft-out 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-rgt-in 99.3%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \sqrt{x} \cdot \sqrt{x + 1}}} \]

      11. rem-square-sqrt 99.4%

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

      12. +-commutative 99.4%

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

      13. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 97.6%

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

4. Final simplification 98.4%

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

Alternative 10: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.66) (+ (pow x -0.5) -1.0) (* (pow x -0.5) (/ 0.5 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.66) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.66d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (x ** (-0.5d0)) * (0.5d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.66) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = Math.pow(x, -0.5) * (0.5 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.66:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = math.pow(x, -0.5) * (0.5 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64((x ^ -0.5) * Float64(0.5 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.66)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (x ^ -0.5) * (0.5 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.66], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.660000000000000031

   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 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. +-commutative 100.0%

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

      2. sub-neg 100.0%

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

      3. fma-udef 100.0%

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

      4. distribute-lft1-in 100.0%

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

      5. metadata-eval 100.0%

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

      6. mul0-lft 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-+r+ 100.0%

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

      9. sub-neg 100.0%

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

      10. neg-sub0 100.0%

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

      11. +-commutative 100.0%

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

      12. sub-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 99.2%

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

   if 0.660000000000000031 < x

   1. Initial program 35.6%

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

      1. flip-- 35.7%

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

      2. add-sqr-sqrt 36.4%

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

      3. +-commutative 36.4%

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

      4. add-sqr-sqrt 37.0%

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

      5. +-commutative 37.0%

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

   6. Applied egg-rr 37.0%

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

      1. *-un-lft-identity 37.0%

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

      2. sqrt-prod 37.0%

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

      3. +-commutative 37.0%

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

      4. times-frac 37.0%

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

      5. pow1/2 37.0%

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

      6. pow-flip 37.0%

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

      7. metadata-eval 37.0%

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

      8. associate--l+ 37.0%

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

   8. Applied egg-rr 37.0%

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

      1. associate-/l/ 37.0%

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

      2. +-commutative 37.0%

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

      3. sub-neg 37.0%

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

      4. associate-+l+ 99.3%

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

      5. neg-mul-1 99.3%

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

      6. *-commutative 99.3%

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

      7. *-rgt-identity 99.3%

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

      8. distribute-lft-out 99.3%

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

      9. metadata-eval 99.3%

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

      10. distribute-rgt-in 99.3%

          \[\leadsto {x}^{-0.5} \cdot \frac{1 + x \cdot 0}{\color{blue}{\sqrt{x + 1} \cdot \sqrt{x + 1} + \sqrt{x} \cdot \sqrt{x + 1}}} \]

      11. rem-square-sqrt 99.4%

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

      12. +-commutative 99.4%

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

      13. +-commutative 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 97.6%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \frac{0.5}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. *-un-lft-identity 63.6%

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

   2. clear-num 63.6%

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

   3. associate-/r/ 63.6%

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

   4. prod-diff 63.6%

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

   5. *-un-lft-identity 63.6%

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

   6. fma-neg 63.6%

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

   7. *-un-lft-identity 63.6%

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

   8. pow1/2 63.6%

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

   9. pow-flip 60.6%

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

   10. metadata-eval 60.6%

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

   11. pow1/2 60.6%

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

   12. pow-flip 63.8%

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

   13. +-commutative 63.8%

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

   14. metadata-eval 63.8%

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

4. Applied egg-rr 63.8%

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

   1. +-commutative 63.8%

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

   2. sub-neg 63.8%

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

   3. fma-udef 63.8%

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

   4. distribute-lft1-in 63.8%

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

   5. metadata-eval 63.8%

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

   6. mul0-lft 63.8%

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

   7. +-commutative 63.8%

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

   8. associate-+r+ 63.8%

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

   9. sub-neg 63.8%

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

   10. neg-sub0 63.8%

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

   11. +-commutative 63.8%

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

   12. sub-neg 63.8%

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

6. Simplified 63.8%

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

7. Taylor expanded in x around 0 44.9%

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

8. Final simplification 44.9%

   \[\leadsto {x}^{-0.5} + -1 \]
9. Add Preprocessing

Alternative 12: 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 63.6%

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

3. Taylor expanded in x around 0 44.6%

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

4. Taylor expanded in x around inf 2.0%

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

5. Final simplification 2.0%

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

Alternative 13: 5.9% 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 63.6%

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

   1. frac-sub 63.6%

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

   2. *-un-lft-identity 63.6%

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

   3. +-commutative 63.6%

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

   4. *-rgt-identity 63.6%

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

   5. sqrt-unprod 63.6%

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

   6. +-commutative 63.6%

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

4. Applied egg-rr 63.6%

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

   1. flip-- 63.6%

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

   2. add-sqr-sqrt 64.0%

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

   3. +-commutative 64.0%

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

   4. add-sqr-sqrt 64.3%

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

   5. +-commutative 64.3%

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

6. Applied egg-rr 64.3%

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

7. Taylor expanded in x around inf 23.2%

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

   1. +-commutative 23.2%

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

9. Simplified 23.2%

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

10. Taylor expanded in x around 0 5.5%

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

11. Final simplification 5.5%

    \[\leadsto 2 \]
12. Add Preprocessing

Developer target: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024011 
(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)))))