2isqrt (example 3.6)

Percentage Accurate: 69.7% → 91.6%

Time: 12.7s

Alternatives: 14

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.7% 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: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 62000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot \left(-1 - x\right)}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 62000000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (/ (/ -1.0 (* x (- -1.0 x))) (* 2.0 (sqrt (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 62000000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = (-1.0 / (x * (-1.0 - x))) / (2.0 * sqrt((1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 62000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = ((-1.0d0) / (x * ((-1.0d0) - x))) / (2.0d0 * sqrt((1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 62000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = (-1.0 / (x * (-1.0 - x))) / (2.0 * Math.sqrt((1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 62000000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = (-1.0 / (x * (-1.0 - x))) / (2.0 * math.sqrt((1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 62000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(Float64(-1.0 / Float64(x * Float64(-1.0 - x))) / Float64(2.0 * sqrt(Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 62000000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = (-1.0 / (x * (-1.0 - x))) / (2.0 * sqrt((1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 62000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.2e7

   1. Initial program 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. clear-num 99.4%

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

      3. associate-/r/ 99.4%

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

      4. prod-diff 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. fma-neg 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. pow1/2 99.4%

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

      9. pow-flip 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-+l- 99.8%

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

      2. expm1-log1p 99.8%

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

      3. expm1-def 99.7%

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

      4. associate--l- 99.7%

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

      5. fma-udef 99.7%

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

      6. distribute-lft1-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. mul0-lft 99.7%

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

      9. metadata-eval 99.7%

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

      10. expm1-def 99.8%

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

      11. expm1-log1p 99.8%

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

   6. Simplified 99.8%

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

   if 6.2e7 < x

   1. Initial program 37.7%

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

      1. flip-- 37.7%

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

      2. frac-times 26.0%

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

      3. metadata-eval 26.0%

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

      4. add-sqr-sqrt 16.1%

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

      5. frac-times 21.4%

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

      6. metadata-eval 21.4%

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

      7. add-sqr-sqrt 37.7%

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

      8. +-commutative 37.7%

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

      9. pow1/2 37.7%

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

      10. pow-flip 37.7%

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

      11. metadata-eval 37.7%

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

      12. inv-pow 37.7%

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

      13. sqrt-pow2 37.7%

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

      14. +-commutative 37.7%

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

      15. metadata-eval 37.7%

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

   4. Applied egg-rr 37.7%

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

      1. frac-2neg 37.7%

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

      2. metadata-eval 37.7%

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

      3. frac-sub 37.7%

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

      4. *-un-lft-identity 37.7%

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

      5. distribute-neg-in 37.7%

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

      6. metadata-eval 37.7%

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

      7. distribute-neg-in 37.7%

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

      8. metadata-eval 37.7%

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

   6. Applied egg-rr 37.7%

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

      1. associate--l+ 80.4%

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

      2. *-commutative 80.4%

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

      3. neg-mul-1 80.4%

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

      4. +-inverses 80.4%

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

      5. metadata-eval 80.4%

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

      6. unsub-neg 80.4%

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

   8. Simplified 80.4%

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

   9. Taylor expanded in x around inf 80.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 62000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot \left(-1 - x\right)}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((1.0 / x) * (-1.0 / (-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) / ((-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 / (-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 / (-1.0 - x))) / (math.pow(x, -0.5) + math.pow((1.0 + x), -0.5))

Julia

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

Wolfram

code[x_] := N[(N[(N[(1.0 / x), $MachinePrecision] * N[(-1.0 / 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 72.7%

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

   1. flip-- 72.6%

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

   2. frac-times 67.4%

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

   3. metadata-eval 67.4%

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

   4. add-sqr-sqrt 63.1%

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

   5. frac-times 65.4%

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

   6. metadata-eval 65.4%

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

   7. add-sqr-sqrt 72.5%

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

   8. +-commutative 72.5%

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

   9. pow1/2 72.5%

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

   10. pow-flip 72.5%

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

   11. metadata-eval 72.5%

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

   12. inv-pow 72.5%

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

   13. sqrt-pow2 72.5%

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

   14. +-commutative 72.5%

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

   15. metadata-eval 72.5%

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

4. Applied egg-rr 72.5%

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

   1. frac-2neg 72.5%

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

   2. metadata-eval 72.5%

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

   3. frac-sub 72.6%

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

   4. *-un-lft-identity 72.6%

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

   5. distribute-neg-in 72.6%

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

   6. metadata-eval 72.6%

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

   7. distribute-neg-in 72.6%

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

   8. metadata-eval 72.6%

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

6. Applied egg-rr 72.6%

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

   1. associate--l+ 91.1%

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

   2. *-commutative 91.1%

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

   3. neg-mul-1 91.1%

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

   4. +-inverses 91.1%

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

   5. metadata-eval 91.1%

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

   6. unsub-neg 91.1%

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

8. Simplified 91.1%

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

   1. metadata-eval 91.1%

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

   2. /-rgt-identity 91.1%

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

   3. sub-neg 91.1%

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

   4. metadata-eval 91.1%

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

   5. distribute-neg-in 91.1%

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

   6. frac-times 91.2%

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

   7. clear-num 91.2%

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

   8. distribute-neg-in 91.2%

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

   9. metadata-eval 91.2%

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

   10. sub-neg 91.2%

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

10. Applied egg-rr 91.2%

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

11. Final simplification 91.2%

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

Alternative 3: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. flip-- 72.6%

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

   2. frac-times 67.4%

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

   3. metadata-eval 67.4%

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

   4. add-sqr-sqrt 63.1%

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

   5. frac-times 65.4%

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

   6. metadata-eval 65.4%

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

   7. add-sqr-sqrt 72.5%

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

   8. +-commutative 72.5%

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

   9. pow1/2 72.5%

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

   10. pow-flip 72.5%

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

   11. metadata-eval 72.5%

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

   12. inv-pow 72.5%

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

   13. sqrt-pow2 72.5%

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

   14. +-commutative 72.5%

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

   15. metadata-eval 72.5%

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

4. Applied egg-rr 72.5%

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

   1. frac-2neg 72.5%

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

   2. metadata-eval 72.5%

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

   3. frac-sub 72.6%

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

   4. *-un-lft-identity 72.6%

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

   5. distribute-neg-in 72.6%

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

   6. metadata-eval 72.6%

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

   7. distribute-neg-in 72.6%

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

   8. metadata-eval 72.6%

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

6. Applied egg-rr 72.6%

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

   1. associate--l+ 91.1%

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

   2. *-commutative 91.1%

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

   3. neg-mul-1 91.1%

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

   4. +-inverses 91.1%

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

   5. metadata-eval 91.1%

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

   6. unsub-neg 91.1%

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

8. Simplified 91.1%

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

   1. expm1-log1p-u 87.6%

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

   2. expm1-udef 68.9%

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

   3. associate-/l/ 68.9%

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

   4. *-commutative 68.9%

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

   5. +-commutative 68.9%

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

10. Applied egg-rr 68.9%

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

    1. expm1-def 87.5%

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

    2. expm1-log1p 91.1%

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

    3. +-commutative 91.1%

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

12. Simplified 91.1%

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

13. Final simplification 91.1%

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

Alternative 4: 91.6% 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 72.7%

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

   1. flip-- 72.6%

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

   2. frac-times 67.4%

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

   3. metadata-eval 67.4%

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

   4. add-sqr-sqrt 63.1%

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

   5. frac-times 65.4%

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

   6. metadata-eval 65.4%

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

   7. add-sqr-sqrt 72.5%

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

   8. +-commutative 72.5%

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

   9. pow1/2 72.5%

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

   10. pow-flip 72.5%

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

   11. metadata-eval 72.5%

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

   12. inv-pow 72.5%

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

   13. sqrt-pow2 72.5%

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

   14. +-commutative 72.5%

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

   15. metadata-eval 72.5%

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

4. Applied egg-rr 72.5%

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

   1. frac-2neg 72.5%

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

   2. metadata-eval 72.5%

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

   3. frac-sub 72.6%

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

   4. *-un-lft-identity 72.6%

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

   5. distribute-neg-in 72.6%

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

   6. metadata-eval 72.6%

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

   7. distribute-neg-in 72.6%

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

   8. metadata-eval 72.6%

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

6. Applied egg-rr 72.6%

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

   1. associate--l+ 91.1%

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

   2. *-commutative 91.1%

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

   3. neg-mul-1 91.1%

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

   4. +-inverses 91.1%

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

   5. metadata-eval 91.1%

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

   6. unsub-neg 91.1%

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

8. Simplified 91.1%

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

9. Final simplification 91.1%

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

Alternative 5: 90.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot \left(-1 - x\right)}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.68)
   (+ -1.0 (+ (pow x -0.5) (* x 0.5)))
   (/ (/ -1.0 (* x (- -1.0 x))) (* 2.0 (sqrt (/ 1.0 x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = -1.0 + (pow(x, -0.5) + (x * 0.5));
	} else {
		tmp = (-1.0 / (x * (-1.0 - x))) / (2.0 * sqrt((1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = (-1.0d0) + ((x ** (-0.5d0)) + (x * 0.5d0))
    else
        tmp = ((-1.0d0) / (x * ((-1.0d0) - x))) / (2.0d0 * sqrt((1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = -1.0 + (Math.pow(x, -0.5) + (x * 0.5));
	} else {
		tmp = (-1.0 / (x * (-1.0 - x))) / (2.0 * Math.sqrt((1.0 / x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.68:
		tmp = -1.0 + (math.pow(x, -0.5) + (x * 0.5))
	else:
		tmp = (-1.0 / (x * (-1.0 - x))) / (2.0 * math.sqrt((1.0 / x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64(-1.0 + Float64((x ^ -0.5) + Float64(x * 0.5)));
	else
		tmp = Float64(Float64(-1.0 / Float64(x * Float64(-1.0 - x))) / Float64(2.0 * sqrt(Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = -1.0 + ((x ^ -0.5) + (x * 0.5));
	else
		tmp = (-1.0 / (x * (-1.0 - x))) / (2.0 * sqrt((1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.68], N[(-1.0 + N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(x * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 100.0%

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

      10. metadata-eval 100.0%

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

      11. pow1/2 100.0%

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

      12. pow-flip 100.0%

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

      13. +-commutative 100.0%

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

      14. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. mul0-lft 100.0%

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

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 99.6%

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

   if 0.680000000000000049 < x

   1. Initial program 38.6%

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

      1. flip-- 38.6%

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

      2. frac-times 27.1%

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

      3. metadata-eval 27.1%

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

      4. add-sqr-sqrt 17.3%

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

      5. frac-times 22.5%

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

      6. metadata-eval 22.5%

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

      7. add-sqr-sqrt 38.6%

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

      8. +-commutative 38.6%

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

      9. pow1/2 38.6%

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

      10. pow-flip 38.6%

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

      11. metadata-eval 38.6%

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

      12. inv-pow 38.6%

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

      13. sqrt-pow2 38.6%

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

      14. +-commutative 38.6%

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

      15. metadata-eval 38.6%

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

   4. Applied egg-rr 38.6%

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

      1. frac-2neg 38.6%

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

      2. metadata-eval 38.6%

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

      3. frac-sub 38.8%

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

      4. *-un-lft-identity 38.8%

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

      5. distribute-neg-in 38.8%

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

      6. metadata-eval 38.8%

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

      7. distribute-neg-in 38.8%

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

      8. metadata-eval 38.8%

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

   6. Applied egg-rr 38.8%

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

      1. associate--l+ 80.8%

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

      2. *-commutative 80.8%

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

      3. neg-mul-1 80.8%

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

      4. +-inverses 80.8%

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

      5. metadata-eval 80.8%

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

      6. unsub-neg 80.8%

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

   8. Simplified 80.8%

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

   9. Taylor expanded in x around inf 79.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x \cdot \left(-1 - x\right)}}{2 \cdot \sqrt{\frac{1}{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{1}{\sqrt{x}} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 - {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8.5e+122)
   (- (/ 1.0 (sqrt x)) (/ 1.0 (+ 1.0 (* x 0.5))))
   (+ 1.0 (- -1.0 (pow x -0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = (1.0 / sqrt(x)) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 1.0 + (-1.0 - pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.5d+122) then
        tmp = (1.0d0 / sqrt(x)) - (1.0d0 / (1.0d0 + (x * 0.5d0)))
    else
        tmp = 1.0d0 + ((-1.0d0) - (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.5e+122) {
		tmp = (1.0 / Math.sqrt(x)) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 1.0 + (-1.0 - Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.5e+122:
		tmp = (1.0 / math.sqrt(x)) - (1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = 1.0 + (-1.0 - math.pow(x, -0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.5e+122)
		tmp = Float64(Float64(1.0 / sqrt(x)) - Float64(1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(1.0 + Float64(-1.0 - (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.5e+122)
		tmp = (1.0 / sqrt(x)) - (1.0 / (1.0 + (x * 0.5)));
	else
		tmp = 1.0 + (-1.0 - (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.5e+122], N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 - N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.50000000000000003e122

   1. Initial program 81.8%

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

   3. Taylor expanded in x around 0 81.2%

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

      1. *-commutative 81.2%

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

   5. Simplified 81.2%

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

   if 8.50000000000000003e122 < x

   1. Initial program 51.9%

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

      1. pow1/2 51.9%

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

      2. pow-to-exp 4.3%

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

      3. +-commutative 4.3%

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

      4. log1p-udef 4.3%

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

   4. Applied egg-rr 4.3%

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

   5. Taylor expanded in x around inf 3.8%

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

      1. mul-1-neg 3.8%

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

   7. Simplified 3.8%

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

      1. inv-pow 3.8%

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

      2. sqrt-pow1 3.8%

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

      3. metadata-eval 3.8%

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

      4. expm1-log1p-u 3.8%

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

      5. expm1-udef 51.9%

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

   9. Applied egg-rr 51.9%

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

       1. expm1-def 3.8%

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

       2. expm1-log1p 3.8%

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

   11. Simplified 3.8%

       \[\leadsto -\color{blue}{{x}^{-0.5}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 3.8%

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

       2. expm1-udef 51.9%

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

       3. log1p-udef 51.9%

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

       4. +-commutative 51.9%

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

       5. add-exp-log 51.9%

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

   13. Applied egg-rr 51.9%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{1}{\sqrt{x}} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 - {x}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. flip-- 72.6%

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

   2. frac-times 67.4%

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

   3. metadata-eval 67.4%

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

   4. add-sqr-sqrt 63.1%

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

   5. frac-times 65.4%

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

   6. metadata-eval 65.4%

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

   7. add-sqr-sqrt 72.5%

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

   8. +-commutative 72.5%

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

   9. pow1/2 72.5%

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

   10. pow-flip 72.5%

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

   11. metadata-eval 72.5%

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

   12. inv-pow 72.5%

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

   13. sqrt-pow2 72.5%

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

   14. +-commutative 72.5%

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

   15. metadata-eval 72.5%

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

4. Applied egg-rr 72.5%

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

   1. frac-2neg 72.5%

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

   2. metadata-eval 72.5%

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

   3. frac-sub 72.6%

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

   4. *-un-lft-identity 72.6%

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

   5. distribute-neg-in 72.6%

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

   6. metadata-eval 72.6%

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

   7. distribute-neg-in 72.6%

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

   8. metadata-eval 72.6%

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

6. Applied egg-rr 72.6%

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

   1. associate--l+ 91.1%

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

   2. *-commutative 91.1%

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

   3. neg-mul-1 91.1%

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

   4. +-inverses 91.1%

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

   5. metadata-eval 91.1%

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

   6. unsub-neg 91.1%

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

8. Simplified 91.1%

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

9. Taylor expanded in x around 0 72.1%

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

10. Final simplification 72.1%

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

Alternative 8: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+76}:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 - {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 8.2e+76)
   (+ -1.0 (+ (pow x -0.5) (* x 0.5)))
   (+ 1.0 (- -1.0 (pow x -0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 8.2e+76) {
		tmp = -1.0 + (pow(x, -0.5) + (x * 0.5));
	} else {
		tmp = 1.0 + (-1.0 - pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.2d+76) then
        tmp = (-1.0d0) + ((x ** (-0.5d0)) + (x * 0.5d0))
    else
        tmp = 1.0d0 + ((-1.0d0) - (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.2e+76) {
		tmp = -1.0 + (Math.pow(x, -0.5) + (x * 0.5));
	} else {
		tmp = 1.0 + (-1.0 - Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.2e+76:
		tmp = -1.0 + (math.pow(x, -0.5) + (x * 0.5))
	else:
		tmp = 1.0 + (-1.0 - math.pow(x, -0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.2e+76)
		tmp = Float64(-1.0 + Float64((x ^ -0.5) + Float64(x * 0.5)));
	else
		tmp = Float64(1.0 + Float64(-1.0 - (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.2e+76)
		tmp = -1.0 + ((x ^ -0.5) + (x * 0.5));
	else
		tmp = 1.0 + (-1.0 - (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.2e+76], N[(-1.0 + N[(N[Power[x, -0.5], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 - N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.1999999999999997e76

   1. Initial program 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. clear-num 89.4%

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

      3. associate-/r/ 89.4%

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

      4. prod-diff 89.4%

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

      5. *-un-lft-identity 89.4%

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

      6. fma-neg 89.4%

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

      7. *-un-lft-identity 89.4%

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

      8. pow1/2 89.4%

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

      9. pow-flip 89.7%

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

      10. metadata-eval 89.7%

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

      11. pow1/2 89.7%

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

      12. pow-flip 89.7%

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

      13. +-commutative 89.7%

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

      14. metadata-eval 89.7%

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

   4. Applied egg-rr 89.7%

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

      1. associate-+l- 89.7%

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

      2. expm1-log1p 89.7%

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

      3. expm1-def 89.9%

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

      4. associate--l- 89.9%

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

      5. fma-udef 89.9%

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

      6. distribute-lft1-in 89.9%

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

      7. metadata-eval 89.9%

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

      8. mul0-lft 89.9%

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

      9. metadata-eval 89.9%

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

      10. expm1-def 89.7%

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

      11. expm1-log1p 89.7%

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

   6. Simplified 89.7%

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

   7. Taylor expanded in x around 0 88.7%

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

   if 8.1999999999999997e76 < x

   1. Initial program 43.9%

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

      1. pow1/2 43.9%

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

      2. pow-to-exp 4.3%

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

      3. +-commutative 4.3%

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

      4. log1p-udef 4.3%

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

   4. Applied egg-rr 4.3%

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

   5. Taylor expanded in x around inf 3.6%

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

      1. mul-1-neg 3.6%

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

   7. Simplified 3.6%

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

      1. inv-pow 3.6%

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

      2. sqrt-pow1 3.6%

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

      3. metadata-eval 3.6%

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

      4. expm1-log1p-u 3.6%

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

      5. expm1-udef 43.9%

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

   9. Applied egg-rr 43.9%

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

       1. expm1-def 3.6%

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

       2. expm1-log1p 3.6%

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

   11. Simplified 3.6%

       \[\leadsto -\color{blue}{{x}^{-0.5}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 3.6%

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

       2. expm1-udef 43.9%

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

       3. log1p-udef 43.9%

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

       4. +-commutative 43.9%

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

       5. add-exp-log 43.9%

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

   13. Applied egg-rr 43.9%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.2 \cdot 10^{+76}:\\ \;\;\;\;-1 + \left({x}^{-0.5} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 - {x}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 - {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.0) (+ -1.0 (pow x -0.5)) (+ 1.0 (- -1.0 (pow x -0.5)))))

C

double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = -1.0 + pow(x, -0.5);
	} else {
		tmp = 1.0 + (-1.0 - pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 4.0d0) then
        tmp = (-1.0d0) + (x ** (-0.5d0))
    else
        tmp = 1.0d0 + ((-1.0d0) - (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 4.0) {
		tmp = -1.0 + Math.pow(x, -0.5);
	} else {
		tmp = 1.0 + (-1.0 - Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 4.0:
		tmp = -1.0 + math.pow(x, -0.5)
	else:
		tmp = 1.0 + (-1.0 - math.pow(x, -0.5))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.0)
		tmp = Float64(-1.0 + (x ^ -0.5));
	else
		tmp = Float64(1.0 + Float64(-1.0 - (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.0)
		tmp = -1.0 + (x ^ -0.5);
	else
		tmp = 1.0 + (-1.0 - (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 4.0], N[(-1.0 + N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 - N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4

   1. Initial program 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. clear-num 99.6%

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

      3. associate-/r/ 99.6%

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

      4. prod-diff 99.6%

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

      5. *-un-lft-identity 99.6%

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

      6. fma-neg 99.6%

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

      7. *-un-lft-identity 99.6%

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

      8. pow1/2 99.6%

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

      9. pow-flip 99.9%

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

      10. metadata-eval 99.9%

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

      11. pow1/2 99.9%

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

      12. pow-flip 99.9%

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

      13. +-commutative 99.9%

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

      14. metadata-eval 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. associate-+l- 99.9%

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

      2. expm1-log1p 99.9%

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

      3. expm1-def 99.9%

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

      4. associate--l- 99.9%

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

      5. fma-udef 99.9%

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

      6. distribute-lft1-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. mul0-lft 99.9%

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

      9. metadata-eval 99.9%

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

      10. expm1-def 99.9%

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

      11. expm1-log1p 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 98.7%

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

   if 4 < x

   1. Initial program 38.0%

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

      1. pow1/2 38.0%

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

      2. pow-to-exp 5.1%

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

      3. +-commutative 5.1%

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

      4. log1p-udef 5.1%

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

   4. Applied egg-rr 5.1%

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

   5. Taylor expanded in x around inf 3.3%

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

      1. mul-1-neg 3.3%

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

   7. Simplified 3.3%

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

      1. inv-pow 3.3%

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

      2. sqrt-pow1 3.3%

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

      3. metadata-eval 3.3%

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

      4. expm1-log1p-u 3.3%

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

      5. expm1-udef 37.4%

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

   9. Applied egg-rr 37.4%

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

       1. expm1-def 3.3%

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

       2. expm1-log1p 3.3%

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

   11. Simplified 3.3%

       \[\leadsto -\color{blue}{{x}^{-0.5}} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 3.3%

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

       2. expm1-udef 37.4%

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

       3. log1p-udef 37.4%

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

       4. +-commutative 37.4%

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

       5. add-exp-log 37.4%

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

   13. Applied egg-rr 37.4%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4:\\ \;\;\;\;-1 + {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-1 - {x}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 53.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. flip-- 72.6%

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

   2. frac-times 67.4%

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

   3. metadata-eval 67.4%

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

   4. add-sqr-sqrt 63.1%

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

   5. frac-times 65.4%

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

   6. metadata-eval 65.4%

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

   7. add-sqr-sqrt 72.5%

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

   8. +-commutative 72.5%

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

   9. pow1/2 72.5%

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

   10. pow-flip 72.5%

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

   11. metadata-eval 72.5%

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

   12. inv-pow 72.5%

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

   13. sqrt-pow2 72.5%

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

   14. +-commutative 72.5%

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

   15. metadata-eval 72.5%

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

4. Applied egg-rr 72.5%

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

5. Taylor expanded in x around 0 57.9%

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

   1. +-commutative 57.9%

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

   2. distribute-lft-in 57.9%

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

   3. pow1 57.9%

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

   4. pow-prod-up 58.0%

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

   5. metadata-eval 58.0%

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

   6. pow1/2 58.0%

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

   7. *-rgt-identity 58.0%

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

7. Applied egg-rr 58.0%

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

8. Final simplification 58.0%

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

Alternative 11: 51.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. *-un-lft-identity 72.7%

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

   2. clear-num 72.7%

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

   3. associate-/r/ 72.7%

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

   4. prod-diff 72.7%

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

   5. *-un-lft-identity 72.7%

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

   6. fma-neg 72.7%

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

   7. *-un-lft-identity 72.7%

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

   8. pow1/2 72.7%

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

   9. pow-flip 68.6%

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

   10. metadata-eval 68.6%

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

   11. pow1/2 68.6%

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

   12. pow-flip 72.9%

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

   13. +-commutative 72.9%

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

   14. metadata-eval 72.9%

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

4. Applied egg-rr 72.9%

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

   1. associate-+l- 72.9%

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

   2. expm1-log1p 72.9%

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

   3. expm1-def 58.6%

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

   4. associate--l- 58.6%

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

   5. fma-udef 58.6%

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

   6. distribute-lft1-in 58.6%

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

   7. metadata-eval 58.6%

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

   8. mul0-lft 58.6%

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

   9. metadata-eval 58.6%

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

   10. expm1-def 72.9%

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

   11. expm1-log1p 72.9%

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

6. Simplified 72.9%

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

7. Taylor expanded in x around 0 56.7%

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

8. Final simplification 56.7%

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

Alternative 12: 2.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. pow1/2 72.7%

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

   2. pow-to-exp 58.2%

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

   3. +-commutative 58.2%

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

   4. log1p-udef 58.2%

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

4. Applied egg-rr 58.2%

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

5. Taylor expanded in x around inf 2.1%

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

   1. mul-1-neg 2.1%

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

7. Simplified 2.1%

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

   1. inv-pow 2.1%

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

   2. sqrt-pow1 2.1%

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

   3. metadata-eval 2.1%

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

   4. expm1-log1p-u 2.1%

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

   5. expm1-udef 17.0%

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

9. Applied egg-rr 17.0%

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

    1. expm1-def 2.1%

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

    2. expm1-log1p 2.1%

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

11. Simplified 2.1%

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

12. Final simplification 2.1%

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

Alternative 13: 51.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.7%

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

   1. *-un-lft-identity 72.7%

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

   2. clear-num 72.7%

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

   3. associate-/r/ 72.7%

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

   4. prod-diff 72.7%

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

   5. *-un-lft-identity 72.7%

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

   6. fma-neg 72.7%

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

   7. *-un-lft-identity 72.7%

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

   8. pow1/2 72.7%

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

   9. pow-flip 68.6%

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

   10. metadata-eval 68.6%

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

   11. pow1/2 68.6%

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

   12. pow-flip 72.9%

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

   13. +-commutative 72.9%

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

   14. metadata-eval 72.9%

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

4. Applied egg-rr 72.9%

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

   1. associate-+l- 72.9%

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

   2. expm1-log1p 72.9%

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

   3. expm1-def 58.6%

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

   4. associate--l- 58.6%

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

   5. fma-udef 58.6%

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

   6. distribute-lft1-in 58.6%

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

   7. metadata-eval 58.6%

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

   8. mul0-lft 58.6%

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

   9. metadata-eval 58.6%

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

   10. expm1-def 72.9%

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

   11. expm1-log1p 72.9%

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

6. Simplified 72.9%

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

8. Applied egg-rr 56.8%

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

   1. associate-*r/ 56.9%

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

   2. *-rgt-identity 56.9%

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

10. Simplified 56.9%

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

11. Taylor expanded in x around inf 56.1%

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

12. Final simplification 56.1%

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

Alternative 14: 1.9% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 72.7%

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

3. Taylor expanded in x around 0 56.4%

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

4. Taylor expanded in x around inf 1.9%

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

5. Final simplification 1.9%

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

Developer target: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024010 
(FPCore (x)
  :name "2isqrt (example 3.6)"
  :precision binary64

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

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