2isqrt (example 3.6)

Percentage Accurate: 39.2% → 99.7%

Time: 10.8s

Alternatives: 15

Speedup: 1.9×

Specification

\[x > 1 \land x < 10^{+308}\]

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: 39.2% 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: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \sqrt{\frac{1}{{x}^{3}}} + 0.5 \cdot \sqrt{\frac{1}{x}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+16)
   (/ (/ 1.0 (+ (pow x -0.5) (pow (+ 1.0 x) -0.5))) (fma x x x))
   (/ (+ (* -0.5 (sqrt (/ 1.0 (pow x 3.0)))) (* 0.5 (sqrt (/ 1.0 x)))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 5e+16) {
		tmp = (1.0 / (pow(x, -0.5) + pow((1.0 + x), -0.5))) / fma(x, x, x);
	} else {
		tmp = ((-0.5 * sqrt((1.0 / pow(x, 3.0)))) + (0.5 * sqrt((1.0 / x)))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+16)
		tmp = Float64(Float64(1.0 / Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5))) / fma(x, x, x));
	else
		tmp = Float64(Float64(Float64(-0.5 * sqrt(Float64(1.0 / (x ^ 3.0)))) + Float64(0.5 * sqrt(Float64(1.0 / x)))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+16], N[(N[(1.0 / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e16

   1. Initial program 57.8%

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

      1. flip-- 57.7%

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

      2. frac-times 56.7%

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

      3. metadata-eval 56.7%

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

      4. add-sqr-sqrt 57.3%

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

      5. frac-times 59.2%

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

      6. metadata-eval 59.2%

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

      7. add-sqr-sqrt 60.3%

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

      8. +-commutative 60.3%

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

      9. inv-pow 60.3%

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

      10. sqrt-pow2 60.3%

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

      11. metadata-eval 60.3%

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

      12. inv-pow 60.3%

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

      13. sqrt-pow2 60.3%

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

      14. +-commutative 60.3%

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

      15. metadata-eval 60.3%

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

   4. Applied egg-rr 60.3%

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

      1. frac-sub 95.2%

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

      2. *-un-lft-identity 95.2%

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

      3. distribute-rgt-in 95.0%

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

      4. *-un-lft-identity 95.0%

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

      5. pow2 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. *-rgt-identity 95.0%

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

      2. associate--l+ 99.3%

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

      3. +-inverses 99.3%

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

      4. metadata-eval 99.3%

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

      5. +-commutative 99.3%

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

      6. unpow2 99.3%

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

      7. fma-define 99.4%

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

   8. Simplified 99.4%

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

      1. div-inv 99.4%

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

      2. add-sqr-sqrt 99.2%

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

      3. associate-*l* 99.4%

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

      4. sqrt-div 99.4%

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

      5. metadata-eval 99.4%

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

      6. fma-undefine 99.3%

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

      7. add-sqr-sqrt 99.3%

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

      8. hypot-define 99.6%

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

      9. sqrt-div 99.5%

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

      10. metadata-eval 99.5%

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

      11. fma-undefine 99.5%

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

      12. add-sqr-sqrt 99.5%

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

      13. hypot-define 99.4%

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

   10. Applied egg-rr 99.4%

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

       1. associate-*l/ 99.4%

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

       2. *-un-lft-identity 99.4%

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

       3. frac-times 99.3%

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

       4. *-un-lft-identity 99.3%

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

       5. hypot-undefine 99.5%

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

       6. add-sqr-sqrt 99.5%

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

       7. hypot-undefine 99.2%

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

       8. add-sqr-sqrt 99.2%

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

       9. add-sqr-sqrt 99.5%

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

       10. fma-undefine 99.7%

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

   12. Applied egg-rr 99.7%

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

   if 5e16 < x

   1. Initial program 42.2%

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

      1. flip-- 42.2%

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

      2. frac-times 20.9%

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

      3. metadata-eval 20.9%

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

      4. add-sqr-sqrt 20.4%

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

      5. frac-times 25.2%

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

      6. metadata-eval 25.2%

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

      7. add-sqr-sqrt 42.2%

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

      8. +-commutative 42.2%

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

      9. inv-pow 42.2%

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

      10. sqrt-pow2 42.2%

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

      11. metadata-eval 42.2%

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

      12. inv-pow 42.2%

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

      13. sqrt-pow2 42.2%

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

      14. +-commutative 42.2%

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

      15. metadata-eval 42.2%

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

   4. Applied egg-rr 42.2%

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

   5. Taylor expanded in x around inf 42.2%

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

      1. *-commutative 42.2%

         \[\leadsto \frac{\frac{1}{x} - \frac{1}{1 + x}}{\color{blue}{\sqrt{\frac{1}{x}} \cdot 2}} \]

      2. unpow1/2 42.2%

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

      3. rem-exp-log 42.2%

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

      4. exp-neg 42.2%

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

      5. exp-prod 42.2%

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

      6. distribute-lft-neg-out 42.2%

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

      7. distribute-rgt-neg-in 42.2%

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

      8. metadata-eval 42.2%

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

      9. exp-to-pow 42.2%

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

   7. Simplified 42.2%

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

   8. Taylor expanded in x around inf 99.8%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}\\ t\_0 \cdot \left(t\_0 \cdot \frac{1}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot x (sqrt x)))))
   (* t_0 (* t_0 (/ 1.0 (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)))))))

C

double code(double x) {
	double t_0 = 1.0 / hypot(x, sqrt(x));
	return t_0 * (t_0 * (1.0 / (pow(x, -0.5) + pow((1.0 + x), -0.5))));
}

Java

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

Python

def code(x):
	t_0 = 1.0 / math.hypot(x, math.sqrt(x))
	return t_0 * (t_0 * (1.0 / (math.pow(x, -0.5) + math.pow((1.0 + x), -0.5))))

Julia

function code(x)
	t_0 = Float64(1.0 / hypot(x, sqrt(x)))
	return Float64(t_0 * Float64(t_0 * Float64(1.0 / Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5)))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / hypot(x, sqrt(x));
	tmp = t_0 * (t_0 * (1.0 / ((x ^ -0.5) + ((1.0 + x) ^ -0.5))));
end

Wolfram

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

Derivation

1. Initial program 43.6%

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

   1. flip-- 43.6%

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

   2. frac-times 24.1%

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

   3. metadata-eval 24.1%

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

   4. add-sqr-sqrt 23.7%

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

   5. frac-times 28.2%

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

   6. metadata-eval 28.2%

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

   7. add-sqr-sqrt 43.8%

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

   8. +-commutative 43.8%

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

   9. inv-pow 43.8%

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

   10. sqrt-pow2 43.8%

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

   11. metadata-eval 43.8%

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

   12. inv-pow 43.8%

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

   13. sqrt-pow2 43.8%

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

   14. +-commutative 43.8%

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

   15. metadata-eval 43.8%

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

4. Applied egg-rr 43.8%

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

   1. frac-sub 46.9%

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

   2. *-un-lft-identity 46.9%

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

   3. distribute-rgt-in 46.9%

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

   4. *-un-lft-identity 46.9%

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

   5. pow2 46.9%

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

6. Applied egg-rr 46.9%

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

   1. *-rgt-identity 46.9%

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

   2. associate--l+ 86.2%

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

   3. +-inverses 86.2%

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

   4. metadata-eval 86.2%

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

   5. +-commutative 86.2%

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

   6. unpow2 86.2%

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

   7. fma-define 86.2%

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

8. Simplified 86.2%

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

   1. div-inv 86.2%

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

   2. add-sqr-sqrt 86.1%

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

   3. associate-*l* 86.1%

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

   4. sqrt-div 86.1%

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

   5. metadata-eval 86.1%

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

   6. fma-undefine 86.1%

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

   7. add-sqr-sqrt 86.1%

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

   8. hypot-define 86.1%

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

   9. sqrt-div 86.1%

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

   10. metadata-eval 86.1%

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

   11. fma-undefine 86.1%

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

   12. add-sqr-sqrt 86.1%

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

   13. hypot-define 99.4%

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

10. Applied egg-rr 99.4%

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

Alternative 3: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}\\ t\_0 \cdot \frac{t\_0}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot x (sqrt x)))))
   (* t_0 (/ t_0 (+ (pow x -0.5) (pow (+ 1.0 x) -0.5))))))

C

double code(double x) {
	double t_0 = 1.0 / hypot(x, sqrt(x));
	return t_0 * (t_0 / (pow(x, -0.5) + pow((1.0 + x), -0.5)));
}

Java

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

Python

def code(x):
	t_0 = 1.0 / math.hypot(x, math.sqrt(x))
	return t_0 * (t_0 / (math.pow(x, -0.5) + math.pow((1.0 + x), -0.5)))

Julia

function code(x)
	t_0 = Float64(1.0 / hypot(x, sqrt(x)))
	return Float64(t_0 * Float64(t_0 / Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / hypot(x, sqrt(x));
	tmp = t_0 * (t_0 / ((x ^ -0.5) + ((1.0 + x) ^ -0.5)));
end

Wolfram

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

Derivation

1. Initial program 43.6%

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

   1. flip-- 43.6%

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

   2. frac-times 24.1%

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

   3. metadata-eval 24.1%

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

   4. add-sqr-sqrt 23.7%

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

   5. frac-times 28.2%

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

   6. metadata-eval 28.2%

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

   7. add-sqr-sqrt 43.8%

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

   8. +-commutative 43.8%

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

   9. inv-pow 43.8%

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

   10. sqrt-pow2 43.8%

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

   11. metadata-eval 43.8%

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

   12. inv-pow 43.8%

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

   13. sqrt-pow2 43.8%

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

   14. +-commutative 43.8%

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

   15. metadata-eval 43.8%

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

4. Applied egg-rr 43.8%

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

   1. frac-sub 46.9%

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

   2. *-un-lft-identity 46.9%

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

   3. distribute-rgt-in 46.9%

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

   4. *-un-lft-identity 46.9%

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

   5. pow2 46.9%

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

6. Applied egg-rr 46.9%

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

   1. *-rgt-identity 46.9%

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

   2. associate--l+ 86.2%

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

   3. +-inverses 86.2%

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

   4. metadata-eval 86.2%

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

   5. +-commutative 86.2%

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

   6. unpow2 86.2%

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

   7. fma-define 86.2%

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

8. Simplified 86.2%

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

   1. add-sqr-sqrt 86.1%

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

   2. associate-/l* 86.1%

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

   3. sqrt-div 86.2%

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

   4. metadata-eval 86.2%

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

   5. fma-undefine 86.1%

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

   6. add-sqr-sqrt 86.1%

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

   7. hypot-define 86.2%

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

   8. sqrt-div 86.1%

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

   9. metadata-eval 86.1%

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

   10. fma-undefine 86.1%

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

   11. add-sqr-sqrt 86.1%

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

   12. hypot-define 99.4%

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

10. Applied egg-rr 99.4%

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

Alternative 4: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}}{\mathsf{hypot}\left({x}^{-0.25}, {\left(1 + x\right)}^{-0.25}\right)}\right)}^{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (/ (/ 1.0 (hypot x (sqrt x))) (hypot (pow x -0.25) (pow (+ 1.0 x) -0.25)))
  2.0))

C

double code(double x) {
	return pow(((1.0 / hypot(x, sqrt(x))) / hypot(pow(x, -0.25), pow((1.0 + x), -0.25))), 2.0);
}

Java

public static double code(double x) {
	return Math.pow(((1.0 / Math.hypot(x, Math.sqrt(x))) / Math.hypot(Math.pow(x, -0.25), Math.pow((1.0 + x), -0.25))), 2.0);
}

Python

def code(x):
	return math.pow(((1.0 / math.hypot(x, math.sqrt(x))) / math.hypot(math.pow(x, -0.25), math.pow((1.0 + x), -0.25))), 2.0)

Julia

function code(x)
	return Float64(Float64(1.0 / hypot(x, sqrt(x))) / hypot((x ^ -0.25), (Float64(1.0 + x) ^ -0.25))) ^ 2.0
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / hypot(x, sqrt(x))) / hypot((x ^ -0.25), ((1.0 + x) ^ -0.25))) ^ 2.0;
end

Wolfram

code[x_] := N[Power[N[(N[(1.0 / N[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Power[x, -0.25], $MachinePrecision] ^ 2 + N[Power[N[(1.0 + x), $MachinePrecision], -0.25], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 43.6%

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

   1. flip-- 43.6%

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

   2. frac-times 24.1%

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

   3. metadata-eval 24.1%

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

   4. add-sqr-sqrt 23.7%

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

   5. frac-times 28.2%

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

   6. metadata-eval 28.2%

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

   7. add-sqr-sqrt 43.8%

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

   8. +-commutative 43.8%

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

   9. inv-pow 43.8%

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

   10. sqrt-pow2 43.8%

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

   11. metadata-eval 43.8%

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

   12. inv-pow 43.8%

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

   13. sqrt-pow2 43.8%

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

   14. +-commutative 43.8%

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

   15. metadata-eval 43.8%

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

4. Applied egg-rr 43.8%

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

   1. frac-sub 46.9%

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

   2. *-un-lft-identity 46.9%

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

   3. distribute-rgt-in 46.9%

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

   4. *-un-lft-identity 46.9%

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

   5. pow2 46.9%

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

6. Applied egg-rr 46.9%

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

   1. *-rgt-identity 46.9%

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

   2. associate--l+ 86.2%

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

   3. +-inverses 86.2%

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

   4. metadata-eval 86.2%

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

   5. +-commutative 86.2%

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

   6. unpow2 86.2%

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

   7. fma-define 86.2%

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

8. Simplified 86.2%

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

   1. add-sqr-sqrt 86.0%

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

   2. add-sqr-sqrt 86.0%

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

   3. times-frac 86.0%

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

10. Applied egg-rr 99.2%

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

    1. unpow2 99.2%

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

12. Simplified 99.2%

    \[\leadsto \color{blue}{{\left(\frac{\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right)}}{\mathsf{hypot}\left({x}^{-0.25}, {\left(1 + x\right)}^{-0.25}\right)}\right)}^{2}} \]
13. Add Preprocessing

Alternative 5: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right) \cdot \mathsf{hypot}\left({x}^{-0.25}, {\left(1 + x\right)}^{-0.25}\right)}\right)}^{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (/ 1.0 (* (hypot x (sqrt x)) (hypot (pow x -0.25) (pow (+ 1.0 x) -0.25))))
  2.0))

C

double code(double x) {
	return pow((1.0 / (hypot(x, sqrt(x)) * hypot(pow(x, -0.25), pow((1.0 + x), -0.25)))), 2.0);
}

Java

public static double code(double x) {
	return Math.pow((1.0 / (Math.hypot(x, Math.sqrt(x)) * Math.hypot(Math.pow(x, -0.25), Math.pow((1.0 + x), -0.25)))), 2.0);
}

Python

def code(x):
	return math.pow((1.0 / (math.hypot(x, math.sqrt(x)) * math.hypot(math.pow(x, -0.25), math.pow((1.0 + x), -0.25)))), 2.0)

Julia

function code(x)
	return Float64(1.0 / Float64(hypot(x, sqrt(x)) * hypot((x ^ -0.25), (Float64(1.0 + x) ^ -0.25)))) ^ 2.0
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (hypot(x, sqrt(x)) * hypot((x ^ -0.25), ((1.0 + x) ^ -0.25)))) ^ 2.0;
end

Wolfram

code[x_] := N[Power[N[(1.0 / N[(N[Sqrt[x ^ 2 + N[Sqrt[x], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[Power[x, -0.25], $MachinePrecision] ^ 2 + N[Power[N[(1.0 + x), $MachinePrecision], -0.25], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]

Derivation

1. Initial program 43.6%

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

   1. flip-- 43.6%

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

   2. frac-times 24.1%

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

   3. metadata-eval 24.1%

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

   4. add-sqr-sqrt 23.7%

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

   5. frac-times 28.2%

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

   6. metadata-eval 28.2%

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

   7. add-sqr-sqrt 43.8%

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

   8. +-commutative 43.8%

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

   9. inv-pow 43.8%

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

   10. sqrt-pow2 43.8%

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

   11. metadata-eval 43.8%

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

   12. inv-pow 43.8%

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

   13. sqrt-pow2 43.8%

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

   14. +-commutative 43.8%

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

   15. metadata-eval 43.8%

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

4. Applied egg-rr 43.8%

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

   1. frac-sub 46.9%

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

   2. *-un-lft-identity 46.9%

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

   3. distribute-rgt-in 46.9%

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

   4. *-un-lft-identity 46.9%

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

   5. pow2 46.9%

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

6. Applied egg-rr 46.9%

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

   1. *-rgt-identity 46.9%

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

   2. associate--l+ 86.2%

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

   3. +-inverses 86.2%

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

   4. metadata-eval 86.2%

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

   5. +-commutative 86.2%

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

   6. unpow2 86.2%

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

   7. fma-define 86.2%

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

8. Simplified 86.2%

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

   1. add-sqr-sqrt 86.0%

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

   2. add-sqr-sqrt 86.0%

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

   3. times-frac 86.0%

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

10. Applied egg-rr 99.2%

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

    1. unpow2 99.2%

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

    2. associate-/l/ 99.1%

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

    3. *-commutative 99.1%

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

12. Simplified 99.1%

    \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(x, \sqrt{x}\right) \cdot \mathsf{hypot}\left({x}^{-0.25}, {\left(1 + x\right)}^{-0.25}\right)}\right)}^{2}} \]
13. Add Preprocessing

Alternative 6: 82.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 10^{-13}:\\ \;\;\;\;\frac{{x}^{-0.5} \cdot 0.5}{\sqrt{x \cdot \left(1 + x\right)}}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 1d-13) then
        tmp = ((x ** (-0.5d0)) * 0.5d0) / sqrt((x * (1.0d0 + x)))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    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((1.0 + x)))) <= 1e-13) {
		tmp = (Math.pow(x, -0.5) * 0.5) / Math.sqrt((x * (1.0 + x)));
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 1e-13)
		tmp = ((x ^ -0.5) * 0.5) / sqrt((x * (1.0 + x)));
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	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[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-13], N[(N[(N[Power[x, -0.5], $MachinePrecision] * 0.5), $MachinePrecision] / N[Sqrt[N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 1e-13

   1. Initial program 41.8%

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

      1. frac-sub 41.8%

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

      2. *-rgt-identity 41.8%

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

      3. *-un-lft-identity 41.8%

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

      4. +-commutative 41.8%

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

      5. sqrt-unprod 41.8%

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

      6. +-commutative 41.8%

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

   4. Applied egg-rr 41.8%

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

   5. Taylor expanded in x around inf 85.0%

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

      1. *-commutative 85.0%

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

      2. unpow1/2 85.0%

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

      3. rem-exp-log 82.2%

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

      4. exp-neg 82.2%

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

      5. exp-prod 82.2%

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

      6. distribute-lft-neg-out 82.2%

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

      7. distribute-rgt-neg-in 82.2%

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

      8. metadata-eval 82.2%

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

      9. exp-to-pow 85.0%

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

   7. Simplified 85.0%

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

   if 1e-13 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 83.9%

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

      1. sub-neg 83.9%

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

      2. inv-pow 83.9%

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

      3. sqrt-pow2 84.8%

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

      4. metadata-eval 84.8%

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

      5. distribute-neg-frac 84.8%

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

      6. metadata-eval 84.8%

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

      7. +-commutative 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. *-rgt-identity 84.8%

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

      2. cancel-sign-sub 84.8%

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

      3. distribute-lft-neg-in 84.8%

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

      4. *-rgt-identity 84.8%

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

      5. distribute-neg-frac 84.8%

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

      6. metadata-eval 84.8%

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

      7. unpow1/2 84.8%

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

      8. exp-to-pow 82.6%

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

      9. log1p-undefine 82.6%

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

      10. *-commutative 82.6%

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

      11. exp-neg 82.7%

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

      12. *-commutative 82.7%

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

      13. distribute-rgt-neg-in 82.7%

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

      14. log1p-undefine 82.7%

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

      15. metadata-eval 82.7%

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

      16. exp-to-pow 85.1%

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

   6. Simplified 85.1%

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

4. Final simplification 85.0%

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

Alternative 7: 82.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 1d-13) then
        tmp = (1.0d0 / (x * (1.0d0 + x))) / ((x ** (-0.5d0)) * 2.0d0)
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    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((1.0 + x)))) <= 1e-13) {
		tmp = (1.0 / (x * (1.0 + x))) / (Math.pow(x, -0.5) * 2.0);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 1e-13)
		tmp = (1.0 / (x * (1.0 + x))) / ((x ^ -0.5) * 2.0);
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	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[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-13], N[(N[(1.0 / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64))))) < 1e-13

   1. Initial program 41.8%

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

      1. flip-- 41.8%

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

      2. frac-times 21.5%

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

      3. metadata-eval 21.5%

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

      4. add-sqr-sqrt 21.0%

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

      5. frac-times 25.7%

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

      6. metadata-eval 25.7%

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

      7. add-sqr-sqrt 41.9%

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

      8. +-commutative 41.9%

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

      9. inv-pow 41.9%

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

      10. sqrt-pow2 41.9%

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

      11. metadata-eval 41.9%

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

      12. inv-pow 41.9%

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

      13. sqrt-pow2 41.9%

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

      14. +-commutative 41.9%

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

      15. metadata-eval 41.9%

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

   4. Applied egg-rr 41.9%

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

      1. frac-sub 44.6%

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

      2. *-un-lft-identity 44.6%

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

      3. distribute-rgt-in 44.6%

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

      4. *-un-lft-identity 44.6%

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

      5. pow2 44.6%

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

   6. Applied egg-rr 44.6%

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

      1. *-rgt-identity 44.6%

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

      2. associate--l+ 85.6%

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

      3. +-inverses 85.6%

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

      4. metadata-eval 85.6%

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

      5. +-commutative 85.6%

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

      6. unpow2 85.6%

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

      7. fma-define 85.6%

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

   8. Simplified 85.6%

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

   9. Taylor expanded in x around inf 84.9%

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

       1. *-commutative 41.9%

          \[\leadsto \frac{\frac{1}{x} - \frac{1}{1 + x}}{\color{blue}{\sqrt{\frac{1}{x}} \cdot 2}} \]

       2. unpow1/2 41.9%

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

       3. rem-exp-log 41.9%

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

       4. exp-neg 41.9%

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

       5. exp-prod 41.9%

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

       6. distribute-lft-neg-out 41.9%

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

       7. distribute-rgt-neg-in 41.9%

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

       8. metadata-eval 41.9%

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

       9. exp-to-pow 41.9%

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

   11. Simplified 84.9%

       \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, x, x\right)}}{\color{blue}{{x}^{-0.5} \cdot 2}} \]
   12. Step-by-step derivation

       1. fma-undefine 85.6%

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

       2. distribute-lft1-in 85.6%

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

       3. +-commutative 85.6%

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

   13. Applied egg-rr 84.9%

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

   if 1e-13 < (-.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 x)) (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 x #s(literal 1 binary64)))))

   1. Initial program 83.9%

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

      1. sub-neg 83.9%

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

      2. inv-pow 83.9%

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

      3. sqrt-pow2 84.8%

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

      4. metadata-eval 84.8%

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

      5. distribute-neg-frac 84.8%

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

      6. metadata-eval 84.8%

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

      7. +-commutative 84.8%

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

   4. Applied egg-rr 84.8%

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

      1. *-rgt-identity 84.8%

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

      2. cancel-sign-sub 84.8%

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

      3. distribute-lft-neg-in 84.8%

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

      4. *-rgt-identity 84.8%

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

      5. distribute-neg-frac 84.8%

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

      6. metadata-eval 84.8%

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

      7. unpow1/2 84.8%

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

      8. exp-to-pow 82.6%

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

      9. log1p-undefine 82.6%

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

      10. *-commutative 82.6%

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

      11. exp-neg 82.7%

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

      12. *-commutative 82.7%

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

      13. distribute-rgt-neg-in 82.7%

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

      14. log1p-undefine 82.7%

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

      15. metadata-eval 82.7%

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

      16. exp-to-pow 85.1%

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

   6. Simplified 85.1%

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

4. Final simplification 84.9%

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

Alternative 8: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.6%

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

   1. frac-sub 43.7%

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

   2. *-rgt-identity 43.7%

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

   3. *-un-lft-identity 43.7%

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

   4. +-commutative 43.7%

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

   5. sqrt-unprod 43.7%

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

   6. +-commutative 43.7%

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

4. Applied egg-rr 43.7%

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

   1. flip-- 44.8%

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

   2. add-sqr-sqrt 45.2%

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

   3. add-sqr-sqrt 46.9%

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

   4. add-sqr-sqrt 46.9%

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

   5. hypot-1-def 46.9%

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

6. Applied egg-rr 46.9%

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

   1. associate--l+ 86.2%

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

   2. +-inverses 86.2%

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

   3. metadata-eval 86.2%

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

   4. +-commutative 86.2%

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

   5. hypot-undefine 86.2%

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

   6. metadata-eval 86.2%

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

   7. rem-square-sqrt 86.2%

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

8. Simplified 86.2%

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

   1. *-un-lft-identity 86.2%

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

   2. associate-/l/ 86.2%

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

   3. distribute-rgt-in 86.2%

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

   4. *-un-lft-identity 86.2%

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

   5. add-sqr-sqrt 86.2%

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

   6. hypot-define 98.8%

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

10. Applied egg-rr 98.8%

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

    1. *-lft-identity 98.8%

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

    2. *-commutative 98.8%

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

12. Simplified 98.8%

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

Alternative 9: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x \cdot \left(1 + \frac{0.5}{x}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+99)
   (/ (/ 1.0 (* x (+ 1.0 x))) (+ (pow x -0.5) (pow (+ 1.0 x) -0.5)))
   (/ (/ 1.0 (+ (sqrt x) (sqrt (+ 1.0 x)))) (* x (+ 1.0 (/ 0.5 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 5e+99) {
		tmp = (1.0 / (x * (1.0 + x))) / (pow(x, -0.5) + pow((1.0 + x), -0.5));
	} else {
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) / (x * (1.0 + (0.5 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5d+99) then
        tmp = (1.0d0 / (x * (1.0d0 + x))) / ((x ** (-0.5d0)) + ((1.0d0 + x) ** (-0.5d0)))
    else
        tmp = (1.0d0 / (sqrt(x) + sqrt((1.0d0 + x)))) / (x * (1.0d0 + (0.5d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e+99) {
		tmp = (1.0 / (x * (1.0 + x))) / (Math.pow(x, -0.5) + Math.pow((1.0 + x), -0.5));
	} else {
		tmp = (1.0 / (Math.sqrt(x) + Math.sqrt((1.0 + x)))) / (x * (1.0 + (0.5 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e+99:
		tmp = (1.0 / (x * (1.0 + x))) / (math.pow(x, -0.5) + math.pow((1.0 + x), -0.5))
	else:
		tmp = (1.0 / (math.sqrt(x) + math.sqrt((1.0 + x)))) / (x * (1.0 + (0.5 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+99)
		tmp = Float64(Float64(1.0 / Float64(x * Float64(1.0 + x))) / Float64((x ^ -0.5) + (Float64(1.0 + x) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / Float64(sqrt(x) + sqrt(Float64(1.0 + x)))) / Float64(x * Float64(1.0 + Float64(0.5 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e+99)
		tmp = (1.0 / (x * (1.0 + x))) / ((x ^ -0.5) + ((1.0 + x) ^ -0.5));
	else
		tmp = (1.0 / (sqrt(x) + sqrt((1.0 + x)))) / (x * (1.0 + (0.5 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e+99], N[(N[(1.0 / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x, -0.5], $MachinePrecision] + N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[x], $MachinePrecision] + N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(1.0 + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000008e99

   1. Initial program 17.7%

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

      1. flip-- 17.7%

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

      2. frac-times 18.3%

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

      3. metadata-eval 18.3%

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

      4. add-sqr-sqrt 18.3%

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

      5. frac-times 18.2%

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

      6. metadata-eval 18.2%

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

      7. add-sqr-sqrt 18.4%

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

      8. +-commutative 18.4%

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

      9. inv-pow 18.4%

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

      10. sqrt-pow2 18.4%

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

      11. metadata-eval 18.4%

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

      12. inv-pow 18.4%

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

      13. sqrt-pow2 18.4%

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

      14. +-commutative 18.4%

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

      15. metadata-eval 18.4%

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

   4. Applied egg-rr 18.4%

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

      1. frac-sub 27.3%

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

      2. *-un-lft-identity 27.3%

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

      3. distribute-rgt-in 27.2%

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

      4. *-un-lft-identity 27.2%

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

      5. pow2 27.2%

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

   6. Applied egg-rr 27.2%

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

      1. *-rgt-identity 27.2%

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

      2. associate--l+ 99.4%

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

      3. +-inverses 99.4%

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

      4. metadata-eval 99.4%

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

      5. +-commutative 99.4%

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

      6. unpow2 99.4%

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

      7. fma-define 99.4%

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

   8. Simplified 99.4%

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

      1. fma-undefine 99.4%

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

      2. distribute-lft1-in 99.4%

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

      3. +-commutative 99.4%

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

   10. Applied egg-rr 99.4%

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

   if 5.00000000000000008e99 < x

   1. Initial program 57.6%

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

      1. frac-sub 57.6%

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

      2. *-rgt-identity 57.6%

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

      3. *-un-lft-identity 57.6%

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

      4. +-commutative 57.6%

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

      5. sqrt-unprod 57.6%

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

      6. +-commutative 57.6%

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

   4. Applied egg-rr 57.6%

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

      1. flip-- 57.6%

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

      2. add-sqr-sqrt 57.4%

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

      3. add-sqr-sqrt 57.6%

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

      4. add-sqr-sqrt 57.6%

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

      5. hypot-1-def 57.6%

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

   6. Applied egg-rr 57.6%

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

      1. associate--l+ 79.2%

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

      2. +-inverses 79.2%

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

      3. metadata-eval 79.2%

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

      4. +-commutative 79.2%

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

      5. hypot-undefine 79.2%

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

      6. metadata-eval 79.2%

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

      7. rem-square-sqrt 79.2%

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

   8. Simplified 79.2%

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

   9. Taylor expanded in x around inf 99.8%

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

       1. associate-*r/ 99.8%

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

       2. metadata-eval 99.8%

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

   11. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} + {\left(1 + x\right)}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{x} + \sqrt{1 + x}}}{x \cdot \left(1 + \frac{0.5}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.6%

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

   1. flip-- 43.6%

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

   2. frac-times 24.1%

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

   3. metadata-eval 24.1%

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

   4. add-sqr-sqrt 23.7%

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

   5. frac-times 28.2%

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

   6. metadata-eval 28.2%

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

   7. add-sqr-sqrt 43.8%

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

   8. +-commutative 43.8%

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

   9. inv-pow 43.8%

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

   10. sqrt-pow2 43.8%

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

   11. metadata-eval 43.8%

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

   12. inv-pow 43.8%

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

   13. sqrt-pow2 43.8%

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

   14. +-commutative 43.8%

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

   15. metadata-eval 43.8%

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

4. Applied egg-rr 43.8%

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

   1. frac-sub 46.9%

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

   2. *-un-lft-identity 46.9%

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

   3. distribute-rgt-in 46.9%

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

   4. *-un-lft-identity 46.9%

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

   5. pow2 46.9%

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

6. Applied egg-rr 46.9%

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

   1. *-rgt-identity 46.9%

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

   2. associate--l+ 86.2%

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

   3. +-inverses 86.2%

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

   4. metadata-eval 86.2%

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

   5. +-commutative 86.2%

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

   6. unpow2 86.2%

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

   7. fma-define 86.2%

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

8. Simplified 86.2%

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

   1. fma-undefine 86.2%

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

   2. distribute-lft1-in 86.2%

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

   3. +-commutative 86.2%

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

10. Applied egg-rr 86.2%

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

11. Final simplification 86.2%

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

Alternative 11: 38.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1}{x}}{1 + {x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.7e+153) (/ (/ 1.0 x) (+ 1.0 (pow x -0.5))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 4.7e+153) {
		tmp = (1.0 / x) / (1.0 + pow(x, -0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.7d+153) then
        tmp = (1.0d0 / x) / (1.0d0 + (x ** (-0.5d0)))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.7e+153) {
		tmp = (1.0 / x) / (1.0 + Math.pow(x, -0.5));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.7e+153:
		tmp = (1.0 / x) / (1.0 + math.pow(x, -0.5))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.7e+153)
		tmp = Float64(Float64(1.0 / x) / Float64(1.0 + (x ^ -0.5)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.7e+153)
		tmp = (1.0 / x) / (1.0 + (x ^ -0.5));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.7e+153], N[(N[(1.0 / x), $MachinePrecision] / N[(1.0 + N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.69999999999999968e153

   1. Initial program 14.1%

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

      1. flip-- 14.1%

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

      2. frac-times 14.4%

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

      3. metadata-eval 14.4%

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

      4. add-sqr-sqrt 14.3%

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

      5. frac-times 14.3%

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

      6. metadata-eval 14.3%

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

      7. add-sqr-sqrt 14.6%

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

      8. +-commutative 14.6%

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

      9. inv-pow 14.6%

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

      10. sqrt-pow2 14.6%

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

      11. metadata-eval 14.6%

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

      12. inv-pow 14.6%

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

      13. sqrt-pow2 14.6%

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

      14. +-commutative 14.6%

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

      15. metadata-eval 14.6%

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

   4. Applied egg-rr 14.6%

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

   5. Taylor expanded in x around 0 7.4%

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

   6. Taylor expanded in x around 0 8.9%

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

   if 4.69999999999999968e153 < x

   1. Initial program 72.6%

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

      1. inv-pow 72.6%

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

      2. add-cube-cbrt 26.4%

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

      3. unpow-prod-down 23.9%

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

      4. fmm-def 4.4%

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

      5. cbrt-prod 4.4%

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

      6. add-sqr-sqrt 4.4%

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

      7. distribute-neg-frac 4.4%

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

      8. metadata-eval 4.4%

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

      9. +-commutative 4.4%

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

   4. Applied egg-rr 4.4%

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

   5. Taylor expanded in x around inf 72.6%

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

      1. distribute-rgt1-in 72.6%

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

      2. metadata-eval 72.6%

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

      3. mul0-lft 72.6%

         \[\leadsto \color{blue}{0} \]

   7. Simplified 72.6%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.7 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{1}{x}}{1 + {x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 81.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} \cdot 2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (/ 1.0 (* x (+ 1.0 x))) (* (pow x -0.5) 2.0)))

C

double code(double x) {
	return (1.0 / (x * (1.0 + x))) / (pow(x, -0.5) * 2.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / (x * (1.0d0 + x))) / ((x ** (-0.5d0)) * 2.0d0)
end function

Java

public static double code(double x) {
	return (1.0 / (x * (1.0 + x))) / (Math.pow(x, -0.5) * 2.0);
}

Python

def code(x):
	return (1.0 / (x * (1.0 + x))) / (math.pow(x, -0.5) * 2.0)

Julia

function code(x)
	return Float64(Float64(1.0 / Float64(x * Float64(1.0 + x))) / Float64((x ^ -0.5) * 2.0))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / (x * (1.0 + x))) / ((x ^ -0.5) * 2.0);
end

Wolfram

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

Derivation

1. Initial program 43.6%

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

   1. flip-- 43.6%

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

   2. frac-times 24.1%

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

   3. metadata-eval 24.1%

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

   4. add-sqr-sqrt 23.7%

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

   5. frac-times 28.2%

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

   6. metadata-eval 28.2%

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

   7. add-sqr-sqrt 43.8%

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

   8. +-commutative 43.8%

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

   9. inv-pow 43.8%

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

   10. sqrt-pow2 43.8%

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

   11. metadata-eval 43.8%

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

   12. inv-pow 43.8%

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

   13. sqrt-pow2 43.8%

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

   14. +-commutative 43.8%

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

   15. metadata-eval 43.8%

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

4. Applied egg-rr 43.8%

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

   1. frac-sub 46.9%

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

   2. *-un-lft-identity 46.9%

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

   3. distribute-rgt-in 46.9%

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

   4. *-un-lft-identity 46.9%

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

   5. pow2 46.9%

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

6. Applied egg-rr 46.9%

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

   1. *-rgt-identity 46.9%

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

   2. associate--l+ 86.2%

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

   3. +-inverses 86.2%

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

   4. metadata-eval 86.2%

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

   5. +-commutative 86.2%

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

   6. unpow2 86.2%

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

   7. fma-define 86.2%

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

8. Simplified 86.2%

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

9. Taylor expanded in x around inf 82.9%

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

    1. *-commutative 41.8%

       \[\leadsto \frac{\frac{1}{x} - \frac{1}{1 + x}}{\color{blue}{\sqrt{\frac{1}{x}} \cdot 2}} \]

    2. unpow1/2 41.8%

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

    3. rem-exp-log 41.8%

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

    4. exp-neg 41.8%

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

    5. exp-prod 41.8%

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

    6. distribute-lft-neg-out 41.8%

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

    7. distribute-rgt-neg-in 41.8%

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

    8. metadata-eval 41.8%

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

    9. exp-to-pow 41.8%

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

11. Simplified 82.9%

    \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(x, x, x\right)}}{\color{blue}{{x}^{-0.5} \cdot 2}} \]
12. Step-by-step derivation

    1. fma-undefine 86.2%

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

    2. distribute-lft1-in 86.2%

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

    3. +-commutative 86.2%

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

13. Applied egg-rr 82.9%

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

14. Final simplification 82.9%

    \[\leadsto \frac{\frac{1}{x \cdot \left(1 + x\right)}}{{x}^{-0.5} \cdot 2} \]
15. Add Preprocessing

Alternative 13: 37.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.12e+123) (* 0.5 (sqrt (/ 1.0 x))) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.12e+123) {
		tmp = 0.5 * sqrt((1.0 / x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.12d+123) then
        tmp = 0.5d0 * sqrt((1.0d0 / x))
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.12e+123) {
		tmp = 0.5 * Math.sqrt((1.0 / x));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.12e+123:
		tmp = 0.5 * math.sqrt((1.0 / x))
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.12e+123)
		tmp = Float64(0.5 * sqrt(Float64(1.0 / x)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.12e+123)
		tmp = 0.5 * sqrt((1.0 / x));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.12e+123], N[(0.5 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.12e123

   1. Initial program 15.8%

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

      1. flip-- 15.8%

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

      2. frac-times 16.2%

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

      3. metadata-eval 16.2%

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

      4. add-sqr-sqrt 16.2%

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

      5. frac-times 16.1%

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

      6. metadata-eval 16.1%

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

      7. add-sqr-sqrt 16.3%

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

      8. +-commutative 16.3%

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

      9. inv-pow 16.3%

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

      10. sqrt-pow2 16.3%

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

      11. metadata-eval 16.3%

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

      12. inv-pow 16.3%

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

      13. sqrt-pow2 16.3%

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

      14. +-commutative 16.3%

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

      15. metadata-eval 16.3%

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

   4. Applied egg-rr 16.3%

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

   5. Taylor expanded in x around 0 7.9%

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

   6. Taylor expanded in x around inf 7.9%

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

      1. *-commutative 7.9%

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

   8. Simplified 7.9%

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

   if 1.12e123 < x

   1. Initial program 63.2%

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

      1. inv-pow 63.2%

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

      2. add-cube-cbrt 23.3%

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

      3. unpow-prod-down 21.1%

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

      4. fmm-def 4.3%

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

      5. cbrt-prod 4.3%

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

      6. add-sqr-sqrt 4.4%

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

      7. distribute-neg-frac 4.4%

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

      8. metadata-eval 4.4%

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

      9. +-commutative 4.4%

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

   4. Applied egg-rr 4.4%

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

   5. Taylor expanded in x around inf 63.2%

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

      1. distribute-rgt1-in 63.2%

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

      2. metadata-eval 63.2%

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

      3. mul0-lft 63.2%

         \[\leadsto \color{blue}{0} \]

   7. Simplified 63.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{+123}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+122}:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 8.2e+122) (pow x -0.5) 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 8.2e+122) {
		tmp = pow(x, -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.2d+122) then
        tmp = x ** (-0.5d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.2e+122) {
		tmp = Math.pow(x, -0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.2e+122:
		tmp = math.pow(x, -0.5)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.2e+122)
		tmp = x ^ -0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.2e+122)
		tmp = x ^ -0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.2e+122], N[Power[x, -0.5], $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.2000000000000004e122

   1. Initial program 15.8%

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

      1. flip-- 15.8%

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

      2. frac-times 16.2%

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

      3. metadata-eval 16.2%

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

      4. add-sqr-sqrt 16.2%

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

      5. frac-times 16.1%

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

      6. metadata-eval 16.1%

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

      7. add-sqr-sqrt 16.3%

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

      8. +-commutative 16.3%

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

      9. inv-pow 16.3%

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

      10. sqrt-pow2 16.3%

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

      11. metadata-eval 16.3%

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

      12. inv-pow 16.3%

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

      13. sqrt-pow2 16.3%

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

      14. +-commutative 16.3%

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

      15. metadata-eval 16.3%

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

   4. Applied egg-rr 16.3%

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

   5. Taylor expanded in x around 0 7.8%

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

      1. unpow1/2 7.8%

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

      2. rem-exp-log 7.8%

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

      3. exp-neg 7.8%

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

      4. exp-prod 7.8%

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

      5. distribute-lft-neg-out 7.8%

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

      6. distribute-rgt-neg-in 7.8%

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

      7. metadata-eval 7.8%

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

      8. exp-to-pow 7.8%

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

   7. Simplified 7.8%

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

   if 8.2000000000000004e122 < x

   1. Initial program 63.2%

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

      1. inv-pow 63.2%

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

      2. add-cube-cbrt 23.3%

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

      3. unpow-prod-down 21.1%

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

      4. fmm-def 4.3%

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

      5. cbrt-prod 4.3%

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

      6. add-sqr-sqrt 4.4%

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

      7. distribute-neg-frac 4.4%

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

      8. metadata-eval 4.4%

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

      9. +-commutative 4.4%

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

   4. Applied egg-rr 4.4%

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

   5. Taylor expanded in x around inf 63.2%

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

      1. distribute-rgt1-in 63.2%

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

      2. metadata-eval 63.2%

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

      3. mul0-lft 63.2%

         \[\leadsto \color{blue}{0} \]

   7. Simplified 63.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 36.2% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 43.6%

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

   1. inv-pow 43.6%

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

   2. add-cube-cbrt 20.0%

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

   3. unpow-prod-down 18.8%

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

   4. fmm-def 9.0%

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

   5. cbrt-prod 8.9%

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

   6. add-sqr-sqrt 9.1%

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

   7. distribute-neg-frac 9.1%

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

   8. metadata-eval 9.1%

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

   9. +-commutative 9.1%

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

4. Applied egg-rr 9.1%

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

5. Taylor expanded in x around inf 38.7%

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

   1. distribute-rgt1-in 38.7%

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

   2. metadata-eval 38.7%

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

   3. mul0-lft 38.7%

      \[\leadsto \color{blue}{0} \]

7. Simplified 38.7%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 98.2% 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 2024152 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64
  :pre (and (> x 1.0) (< x 1e+308))

  :alt
  (! :herbie-platform default (/ 1 (+ (* (+ x 1) (sqrt x)) (* x (sqrt (+ x 1))))))

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