2isqrt (example 3.6)

Percentage Accurate: 68.7% → 99.6%

Time: 11.3s

Alternatives: 8

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.7% 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: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, {x}^{-1.5} \cdot \frac{1.5}{x}, {x}^{-1.5} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-15)
   (fma -0.25 (* (pow x -1.5) (/ 1.5 x)) (* (pow x -1.5) 0.5))
   (- (pow x -0.5) (cbrt (pow (+ 1.0 x) -1.5)))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-15) {
		tmp = fma(-0.25, (pow(x, -1.5) * (1.5 / x)), (pow(x, -1.5) * 0.5));
	} else {
		tmp = pow(x, -0.5) - cbrt(pow((1.0 + x), -1.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-15)
		tmp = fma(-0.25, Float64((x ^ -1.5) * Float64(1.5 / x)), Float64((x ^ -1.5) * 0.5));
	else
		tmp = Float64((x ^ -0.5) - cbrt((Float64(1.0 + x) ^ -1.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-15], N[(-0.25 * N[(N[Power[x, -1.5], $MachinePrecision] * N[(1.5 / x), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[Power[N[(1.0 + x), $MachinePrecision], -1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 2.0000000000000002e-15

   1. Initial program 42.4%

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

      1. flip-- 42.4%

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

      2. frac-times 25.8%

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

      3. metadata-eval 25.8%

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

      4. add-sqr-sqrt 21.3%

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

      5. frac-times 27.5%

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

      6. metadata-eval 27.5%

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

      7. add-sqr-sqrt 42.5%

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

      8. +-commutative 42.5%

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

      9. pow1/2 42.5%

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

      10. pow-flip 42.5%

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

      11. metadata-eval 42.5%

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

      12. inv-pow 42.5%

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

      13. sqrt-pow2 42.5%

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

      14. +-commutative 42.5%

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

      15. metadata-eval 42.5%

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

   4. Applied egg-rr 42.5%

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

      1. frac-sub 43.1%

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

      2. associate-/r* 43.1%

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

      3. *-un-lft-identity 43.1%

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

      4. +-commutative 43.1%

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

      5. *-rgt-identity 43.1%

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

      6. associate--l+ 43.1%

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

      7. +-commutative 43.1%

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

   6. Applied egg-rr 43.1%

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

   7. Taylor expanded in x around inf 66.3%

      \[\leadsto \color{blue}{-0.25 \cdot \frac{-0.5 \cdot \sqrt{\frac{1}{x}} + 2 \cdot \sqrt{\frac{1}{x}}}{{x}^{2}} + 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]

   9. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.25, {x}^{-1.5} \cdot \frac{1.5}{x}, 0.5 \cdot {x}^{-1.5}\right)} \]

   if 2.0000000000000002e-15 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-+l- 99.8%

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

      2. expm1-log1p 99.8%

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

      3. expm1-def 99.7%

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

      4. associate--l- 99.7%

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

      5. fma-udef 99.7%

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

      6. distribute-lft1-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. mul0-lft 99.7%

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

      9. metadata-eval 99.7%

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

      10. expm1-def 99.8%

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

      11. expm1-log1p 99.8%

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

   6. Simplified 99.8%

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

      1. metadata-eval 99.8%

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

      2. pow-flip 99.8%

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

      3. +-commutative 99.8%

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

      4. pow1/2 99.8%

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

      5. add-cbrt-cube 99.8%

         \[\leadsto {x}^{-0.5} - \color{blue}{\sqrt[3]{\left(\frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \frac{1}{\sqrt{x + 1}}}} \]

      6. pow3 99.8%

         \[\leadsto {x}^{-0.5} - \sqrt[3]{\color{blue}{{\left(\frac{1}{\sqrt{x + 1}}\right)}^{3}}} \]

      7. pow1/2 99.8%

         \[\leadsto {x}^{-0.5} - \sqrt[3]{{\left(\frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right)}^{3}} \]

      8. +-commutative 99.8%

         \[\leadsto {x}^{-0.5} - \sqrt[3]{{\left(\frac{1}{{\color{blue}{\left(1 + x\right)}}^{0.5}}\right)}^{3}} \]

      9. pow-flip 99.8%

         \[\leadsto {x}^{-0.5} - \sqrt[3]{{\color{blue}{\left({\left(1 + x\right)}^{\left(-0.5\right)}\right)}}^{3}} \]

      10. metadata-eval 99.8%

          \[\leadsto {x}^{-0.5} - \sqrt[3]{{\left({\left(1 + x\right)}^{\color{blue}{-0.5}}\right)}^{3}} \]

      11. pow-pow 99.8%

          \[\leadsto {x}^{-0.5} - \sqrt[3]{\color{blue}{{\left(1 + x\right)}^{\left(-0.5 \cdot 3\right)}}} \]

      12. +-commutative 99.8%

          \[\leadsto {x}^{-0.5} - \sqrt[3]{{\color{blue}{\left(x + 1\right)}}^{\left(-0.5 \cdot 3\right)}} \]

      13. metadata-eval 99.8%

          \[\leadsto {x}^{-0.5} - \sqrt[3]{{\left(x + 1\right)}^{\color{blue}{-1.5}}} \]

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, {x}^{-1.5} \cdot \frac{1.5}{x}, {x}^{-1.5} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-15)
   (* (pow x -1.5) 0.5)
   (- (pow x -0.5) (cbrt (pow (+ 1.0 x) -1.5)))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-15) {
		tmp = pow(x, -1.5) * 0.5;
	} else {
		tmp = pow(x, -0.5) - cbrt(pow((1.0 + x), -1.5));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-15) {
		tmp = Math.pow(x, -1.5) * 0.5;
	} else {
		tmp = Math.pow(x, -0.5) - Math.cbrt(Math.pow((1.0 + x), -1.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-15)
		tmp = Float64((x ^ -1.5) * 0.5);
	else
		tmp = Float64((x ^ -0.5) - cbrt((Float64(1.0 + x) ^ -1.5)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-15], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[Power[N[(1.0 + x), $MachinePrecision], -1.5], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 2.0000000000000002e-15

   1. Initial program 42.4%

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

      1. flip-- 42.4%

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

      2. frac-times 25.8%

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

      3. metadata-eval 25.8%

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

      4. add-sqr-sqrt 21.3%

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

      5. frac-times 27.5%

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

      6. metadata-eval 27.5%

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

      7. add-sqr-sqrt 42.5%

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

      8. +-commutative 42.5%

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

      9. pow1/2 42.5%

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

      10. pow-flip 42.5%

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

      11. metadata-eval 42.5%

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

      12. inv-pow 42.5%

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

      13. sqrt-pow2 42.5%

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

      14. +-commutative 42.5%

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

      15. metadata-eval 42.5%

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

   4. Applied egg-rr 42.5%

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

      1. frac-sub 43.1%

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

      2. associate-/r* 43.1%

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

      3. *-un-lft-identity 43.1%

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

      4. +-commutative 43.1%

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

      5. *-rgt-identity 43.1%

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

      6. associate--l+ 43.1%

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

      7. +-commutative 43.1%

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

   6. Applied egg-rr 43.1%

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

   7. Taylor expanded in x around inf 65.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   8. Step-by-step derivation

      1. unpow-1 65.9%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{\left({x}^{3}\right)}^{-1}}} \]

      2. exp-to-pow 63.9%

         \[\leadsto 0.5 \cdot \sqrt{{\color{blue}{\left(e^{\log x \cdot 3}\right)}}^{-1}} \]

      3. *-commutative 63.9%

         \[\leadsto 0.5 \cdot \sqrt{{\left(e^{\color{blue}{3 \cdot \log x}}\right)}^{-1}} \]

      4. exp-prod 64.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\left(3 \cdot \log x\right) \cdot -1}}} \]

      5. *-commutative 64.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \left(3 \cdot \log x\right)}}} \]

      6. associate-*r* 64.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

      7. metadata-eval 64.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

      8. *-commutative 64.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

      9. exp-to-pow 66.1%

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

      10. metadata-eval 66.1%

          \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

      11. pow-sqr 66.2%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-1.5} \cdot {x}^{-1.5}}} \]

      12. rem-sqrt-square 99.6%

          \[\leadsto 0.5 \cdot \color{blue}{\left|{x}^{-1.5}\right|} \]

      13. rem-square-sqrt 99.2%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

      14. fabs-sqr 99.2%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}\right)} \]

      15. rem-square-sqrt 99.6%

          \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

   9. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5}} \]

   if 2.0000000000000002e-15 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-+l- 99.8%

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

      2. expm1-log1p 99.8%

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

      3. expm1-def 99.7%

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

      4. associate--l- 99.7%

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

      5. fma-udef 99.7%

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

      6. distribute-lft1-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. mul0-lft 99.7%

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

      9. metadata-eval 99.7%

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

      10. expm1-def 99.8%

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

      11. expm1-log1p 99.8%

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

   6. Simplified 99.8%

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

      1. metadata-eval 99.8%

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

      2. pow-flip 99.8%

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

      3. +-commutative 99.8%

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

      4. pow1/2 99.8%

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

      5. add-cbrt-cube 99.8%

         \[\leadsto {x}^{-0.5} - \color{blue}{\sqrt[3]{\left(\frac{1}{\sqrt{x + 1}} \cdot \frac{1}{\sqrt{x + 1}}\right) \cdot \frac{1}{\sqrt{x + 1}}}} \]

      6. pow3 99.8%

         \[\leadsto {x}^{-0.5} - \sqrt[3]{\color{blue}{{\left(\frac{1}{\sqrt{x + 1}}\right)}^{3}}} \]

      7. pow1/2 99.8%

         \[\leadsto {x}^{-0.5} - \sqrt[3]{{\left(\frac{1}{\color{blue}{{\left(x + 1\right)}^{0.5}}}\right)}^{3}} \]

      8. +-commutative 99.8%

         \[\leadsto {x}^{-0.5} - \sqrt[3]{{\left(\frac{1}{{\color{blue}{\left(1 + x\right)}}^{0.5}}\right)}^{3}} \]

      9. pow-flip 99.8%

         \[\leadsto {x}^{-0.5} - \sqrt[3]{{\color{blue}{\left({\left(1 + x\right)}^{\left(-0.5\right)}\right)}}^{3}} \]

      10. metadata-eval 99.8%

          \[\leadsto {x}^{-0.5} - \sqrt[3]{{\left({\left(1 + x\right)}^{\color{blue}{-0.5}}\right)}^{3}} \]

      11. pow-pow 99.8%

          \[\leadsto {x}^{-0.5} - \sqrt[3]{\color{blue}{{\left(1 + x\right)}^{\left(-0.5 \cdot 3\right)}}} \]

      12. +-commutative 99.8%

          \[\leadsto {x}^{-0.5} - \sqrt[3]{{\color{blue}{\left(x + 1\right)}}^{\left(-0.5 \cdot 3\right)}} \]

      13. metadata-eval 99.8%

          \[\leadsto {x}^{-0.5} - \sqrt[3]{{\left(x + 1\right)}^{\color{blue}{-1.5}}} \]

   8. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - \sqrt[3]{{\left(1 + x\right)}^{-1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 2e-15)
   (* (pow x -1.5) 0.5)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-15) {
		tmp = pow(x, -1.5) * 0.5;
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 2d-15) then
        tmp = (x ** (-1.5d0)) * 0.5d0
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 2e-15) {
		tmp = Math.pow(x, -1.5) * 0.5;
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 2e-15:
		tmp = math.pow(x, -1.5) * 0.5
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 2e-15)
		tmp = Float64((x ^ -1.5) * 0.5);
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 2e-15)
		tmp = (x ^ -1.5) * 0.5;
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-15], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 2.0000000000000002e-15

   1. Initial program 42.4%

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

      1. flip-- 42.4%

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

      2. frac-times 25.8%

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

      3. metadata-eval 25.8%

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

      4. add-sqr-sqrt 21.3%

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

      5. frac-times 27.5%

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

      6. metadata-eval 27.5%

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

      7. add-sqr-sqrt 42.5%

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

      8. +-commutative 42.5%

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

      9. pow1/2 42.5%

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

      10. pow-flip 42.5%

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

      11. metadata-eval 42.5%

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

      12. inv-pow 42.5%

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

      13. sqrt-pow2 42.5%

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

      14. +-commutative 42.5%

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

      15. metadata-eval 42.5%

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

   4. Applied egg-rr 42.5%

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

      1. frac-sub 43.1%

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

      2. associate-/r* 43.1%

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

      3. *-un-lft-identity 43.1%

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

      4. +-commutative 43.1%

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

      5. *-rgt-identity 43.1%

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

      6. associate--l+ 43.1%

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

      7. +-commutative 43.1%

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

   6. Applied egg-rr 43.1%

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

   7. Taylor expanded in x around inf 65.9%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   8. Step-by-step derivation

      1. unpow-1 65.9%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{\left({x}^{3}\right)}^{-1}}} \]

      2. exp-to-pow 63.9%

         \[\leadsto 0.5 \cdot \sqrt{{\color{blue}{\left(e^{\log x \cdot 3}\right)}}^{-1}} \]

      3. *-commutative 63.9%

         \[\leadsto 0.5 \cdot \sqrt{{\left(e^{\color{blue}{3 \cdot \log x}}\right)}^{-1}} \]

      4. exp-prod 64.1%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\left(3 \cdot \log x\right) \cdot -1}}} \]

      5. *-commutative 64.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \left(3 \cdot \log x\right)}}} \]

      6. associate-*r* 64.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

      7. metadata-eval 64.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

      8. *-commutative 64.1%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

      9. exp-to-pow 66.1%

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

      10. metadata-eval 66.1%

          \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

      11. pow-sqr 66.2%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-1.5} \cdot {x}^{-1.5}}} \]

      12. rem-sqrt-square 99.6%

          \[\leadsto 0.5 \cdot \color{blue}{\left|{x}^{-1.5}\right|} \]

      13. rem-square-sqrt 99.2%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

      14. fabs-sqr 99.2%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}\right)} \]

      15. rem-square-sqrt 99.6%

          \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

   9. Simplified 99.6%

      \[\leadsto \color{blue}{0.5 \cdot {x}^{-1.5}} \]

   if 2.0000000000000002e-15 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. clear-num 99.5%

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

      3. associate-/r/ 99.5%

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

      4. prod-diff 99.5%

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

      5. *-un-lft-identity 99.5%

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

      6. fma-neg 99.5%

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

      7. *-un-lft-identity 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 99.8%

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

      10. metadata-eval 99.8%

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

      11. pow1/2 99.8%

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

      12. pow-flip 99.8%

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

      13. +-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-+l- 99.8%

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

      2. expm1-log1p 99.8%

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

      3. expm1-def 99.7%

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

      4. associate--l- 99.7%

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

      5. fma-udef 99.7%

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

      6. distribute-lft1-in 99.7%

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

      7. metadata-eval 99.7%

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

      8. mul0-lft 99.7%

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

      9. metadata-eval 99.7%

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

      10. expm1-def 99.8%

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

      11. expm1-log1p 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-15}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. pow1/2 99.7%

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

      9. pow-flip 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 1 < x

   1. Initial program 43.0%

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

      1. flip-- 43.0%

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

      2. frac-times 26.6%

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

      3. metadata-eval 26.6%

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

      4. add-sqr-sqrt 22.3%

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

      5. frac-times 28.4%

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

      6. metadata-eval 28.4%

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

      7. add-sqr-sqrt 43.2%

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

      8. +-commutative 43.2%

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

      9. pow1/2 43.2%

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

      10. pow-flip 43.2%

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

      11. metadata-eval 43.2%

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

      12. inv-pow 43.2%

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

      13. sqrt-pow2 43.2%

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

      14. +-commutative 43.2%

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

      15. metadata-eval 43.2%

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

   4. Applied egg-rr 43.2%

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

      1. frac-sub 43.9%

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

      2. associate-/r* 44.0%

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

      3. *-un-lft-identity 44.0%

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

      4. +-commutative 44.0%

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

      5. *-rgt-identity 44.0%

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

      6. associate--l+ 44.0%

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

      7. +-commutative 44.0%

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

   6. Applied egg-rr 44.0%

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

   7. Taylor expanded in x around inf 65.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   8. Step-by-step derivation

      1. unpow-1 65.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{\left({x}^{3}\right)}^{-1}}} \]

      2. exp-to-pow 63.5%

         \[\leadsto 0.5 \cdot \sqrt{{\color{blue}{\left(e^{\log x \cdot 3}\right)}}^{-1}} \]

      3. *-commutative 63.5%

         \[\leadsto 0.5 \cdot \sqrt{{\left(e^{\color{blue}{3 \cdot \log x}}\right)}^{-1}} \]

      4. exp-prod 63.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\left(3 \cdot \log x\right) \cdot -1}}} \]

      5. *-commutative 63.7%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \left(3 \cdot \log x\right)}}} \]

      6. associate-*r* 63.7%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

      7. metadata-eval 63.7%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

      8. *-commutative 63.7%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

      9. exp-to-pow 65.7%

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

      10. metadata-eval 65.7%

          \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

      11. pow-sqr 65.7%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-1.5} \cdot {x}^{-1.5}}} \]

      12. rem-sqrt-square 98.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left|{x}^{-1.5}\right|} \]

      13. rem-square-sqrt 98.3%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

      14. fabs-sqr 98.3%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}\right)} \]

      15. rem-square-sqrt 98.7%

          \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

   9. Simplified 98.7%

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

4. Final simplification 99.1%

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

Alternative 5: 96.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5) (pow x -0.5) (* (pow x -1.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = pow(x, -0.5);
	} else {
		tmp = pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = x ** (-0.5d0)
    else
        tmp = (x ** (-1.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = math.pow(x, -0.5)
	else:
		tmp = math.pow(x, -1.5) * 0.5
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.5)
		tmp = x ^ -0.5;
	else
		tmp = (x ^ -1.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[Power[x, -0.5], $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.7%

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

      1. inv-pow 99.7%

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

      2. add-sqr-sqrt 99.2%

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

      3. unpow-prod-down 98.9%

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

      4. pow1/2 98.9%

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

      5. sqrt-pow1 99.1%

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

      6. metadata-eval 99.1%

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

      7. pow1/2 99.1%

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

      8. sqrt-pow1 99.0%

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

      9. metadata-eval 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. pow-sqr 99.2%

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

      2. metadata-eval 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in x around inf 96.3%

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

      1. inv-pow 96.3%

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

      2. sqrt-pow1 96.5%

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

      3. metadata-eval 96.5%

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

      4. expm1-log1p-u 89.0%

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

      5. expm1-udef 89.0%

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

   9. Applied egg-rr 89.0%

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

       1. expm1-def 89.0%

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

       2. expm1-log1p 96.5%

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

   11. Simplified 96.5%

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

   if 0.5 < x

   1. Initial program 43.0%

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

      1. flip-- 43.0%

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

      2. frac-times 26.6%

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

      3. metadata-eval 26.6%

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

      4. add-sqr-sqrt 22.3%

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

      5. frac-times 28.4%

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

      6. metadata-eval 28.4%

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

      7. add-sqr-sqrt 43.2%

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

      8. +-commutative 43.2%

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

      9. pow1/2 43.2%

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

      10. pow-flip 43.2%

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

      11. metadata-eval 43.2%

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

      12. inv-pow 43.2%

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

      13. sqrt-pow2 43.2%

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

      14. +-commutative 43.2%

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

      15. metadata-eval 43.2%

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

   4. Applied egg-rr 43.2%

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

      1. frac-sub 43.9%

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

      2. associate-/r* 44.0%

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

      3. *-un-lft-identity 44.0%

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

      4. +-commutative 44.0%

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

      5. *-rgt-identity 44.0%

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

      6. associate--l+ 44.0%

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

      7. +-commutative 44.0%

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

   6. Applied egg-rr 44.0%

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

   7. Taylor expanded in x around inf 65.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   8. Step-by-step derivation

      1. unpow-1 65.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{\left({x}^{3}\right)}^{-1}}} \]

      2. exp-to-pow 63.5%

         \[\leadsto 0.5 \cdot \sqrt{{\color{blue}{\left(e^{\log x \cdot 3}\right)}}^{-1}} \]

      3. *-commutative 63.5%

         \[\leadsto 0.5 \cdot \sqrt{{\left(e^{\color{blue}{3 \cdot \log x}}\right)}^{-1}} \]

      4. exp-prod 63.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\left(3 \cdot \log x\right) \cdot -1}}} \]

      5. *-commutative 63.7%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \left(3 \cdot \log x\right)}}} \]

      6. associate-*r* 63.7%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

      7. metadata-eval 63.7%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

      8. *-commutative 63.7%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

      9. exp-to-pow 65.7%

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

      10. metadata-eval 65.7%

          \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

      11. pow-sqr 65.7%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-1.5} \cdot {x}^{-1.5}}} \]

      12. rem-sqrt-square 98.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left|{x}^{-1.5}\right|} \]

      13. rem-square-sqrt 98.3%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

      14. fabs-sqr 98.3%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}\right)} \]

      15. rem-square-sqrt 98.7%

          \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

   9. Simplified 98.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;{x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.68) (+ (pow x -0.5) -1.0) (* (pow x -1.5) 0.5)))

C

double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = pow(x, -0.5) + -1.0;
	} else {
		tmp = pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.68d0) then
        tmp = (x ** (-0.5d0)) + (-1.0d0)
    else
        tmp = (x ** (-1.5d0)) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.68) {
		tmp = Math.pow(x, -0.5) + -1.0;
	} else {
		tmp = Math.pow(x, -1.5) * 0.5;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.68:
		tmp = math.pow(x, -0.5) + -1.0
	else:
		tmp = math.pow(x, -1.5) * 0.5
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.68)
		tmp = Float64((x ^ -0.5) + -1.0);
	else
		tmp = Float64((x ^ -1.5) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.68)
		tmp = (x ^ -0.5) + -1.0;
	else
		tmp = (x ^ -1.5) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.68], N[(N[Power[x, -0.5], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Power[x, -1.5], $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. clear-num 99.7%

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

      3. associate-/r/ 99.7%

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

      4. prod-diff 99.7%

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

      5. *-un-lft-identity 99.7%

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

      6. fma-neg 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. pow1/2 99.7%

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

      9. pow-flip 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 99.2%

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

   if 0.680000000000000049 < x

   1. Initial program 43.0%

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

      1. flip-- 43.0%

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

      2. frac-times 26.6%

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

      3. metadata-eval 26.6%

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

      4. add-sqr-sqrt 22.3%

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

      5. frac-times 28.4%

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

      6. metadata-eval 28.4%

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

      7. add-sqr-sqrt 43.2%

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

      8. +-commutative 43.2%

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

      9. pow1/2 43.2%

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

      10. pow-flip 43.2%

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

      11. metadata-eval 43.2%

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

      12. inv-pow 43.2%

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

      13. sqrt-pow2 43.2%

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

      14. +-commutative 43.2%

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

      15. metadata-eval 43.2%

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

   4. Applied egg-rr 43.2%

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

      1. frac-sub 43.9%

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

      2. associate-/r* 44.0%

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

      3. *-un-lft-identity 44.0%

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

      4. +-commutative 44.0%

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

      5. *-rgt-identity 44.0%

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

      6. associate--l+ 44.0%

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

      7. +-commutative 44.0%

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

   6. Applied egg-rr 44.0%

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

   7. Taylor expanded in x around inf 65.5%

      \[\leadsto \color{blue}{0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   8. Step-by-step derivation

      1. unpow-1 65.5%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{\left({x}^{3}\right)}^{-1}}} \]

      2. exp-to-pow 63.5%

         \[\leadsto 0.5 \cdot \sqrt{{\color{blue}{\left(e^{\log x \cdot 3}\right)}}^{-1}} \]

      3. *-commutative 63.5%

         \[\leadsto 0.5 \cdot \sqrt{{\left(e^{\color{blue}{3 \cdot \log x}}\right)}^{-1}} \]

      4. exp-prod 63.7%

         \[\leadsto 0.5 \cdot \sqrt{\color{blue}{e^{\left(3 \cdot \log x\right) \cdot -1}}} \]

      5. *-commutative 63.7%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-1 \cdot \left(3 \cdot \log x\right)}}} \]

      6. associate-*r* 63.7%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\left(-1 \cdot 3\right) \cdot \log x}}} \]

      7. metadata-eval 63.7%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{-3} \cdot \log x}} \]

      8. *-commutative 63.7%

         \[\leadsto 0.5 \cdot \sqrt{e^{\color{blue}{\log x \cdot -3}}} \]

      9. exp-to-pow 65.7%

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

      10. metadata-eval 65.7%

          \[\leadsto 0.5 \cdot \sqrt{{x}^{\color{blue}{\left(2 \cdot -1.5\right)}}} \]

      11. pow-sqr 65.7%

          \[\leadsto 0.5 \cdot \sqrt{\color{blue}{{x}^{-1.5} \cdot {x}^{-1.5}}} \]

      12. rem-sqrt-square 98.7%

          \[\leadsto 0.5 \cdot \color{blue}{\left|{x}^{-1.5}\right|} \]

      13. rem-square-sqrt 98.3%

          \[\leadsto 0.5 \cdot \left|\color{blue}{\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}}\right| \]

      14. fabs-sqr 98.3%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{{x}^{-1.5}} \cdot \sqrt{{x}^{-1.5}}\right)} \]

      15. rem-square-sqrt 98.7%

          \[\leadsto 0.5 \cdot \color{blue}{{x}^{-1.5}} \]

   9. Simplified 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.68:\\ \;\;\;\;{x}^{-0.5} + -1\\ \mathbf{else}:\\ \;\;\;\;{x}^{-1.5} \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return x ^ -0.5
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. inv-pow 68.9%

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

   2. add-sqr-sqrt 61.0%

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

   3. unpow-prod-down 55.5%

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

   4. pow1/2 55.5%

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

   5. sqrt-pow1 55.8%

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

   6. metadata-eval 55.8%

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

   7. pow1/2 55.8%

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

   8. sqrt-pow1 54.9%

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

   9. metadata-eval 54.9%

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

4. Applied egg-rr 54.9%

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

   1. pow-sqr 56.1%

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

   2. metadata-eval 56.1%

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

6. Simplified 56.1%

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

7. Taylor expanded in x around inf 47.0%

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

   1. inv-pow 47.0%

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

   2. sqrt-pow1 47.0%

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

   3. metadata-eval 47.0%

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

   4. expm1-log1p-u 43.6%

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

   5. expm1-udef 63.3%

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

9. Applied egg-rr 63.3%

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

    1. expm1-def 43.6%

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

    2. expm1-log1p 47.0%

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

11. Simplified 47.0%

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

12. Final simplification 47.0%

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

Alternative 8: 2.4% accurate, 41.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return (x * 0.5) + -1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. inv-pow 68.9%

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

   2. add-sqr-sqrt 61.0%

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

   3. unpow-prod-down 55.5%

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

   4. pow1/2 55.5%

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

   5. sqrt-pow1 55.8%

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

   6. metadata-eval 55.8%

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

   7. pow1/2 55.8%

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

   8. sqrt-pow1 54.9%

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

   9. metadata-eval 54.9%

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

4. Applied egg-rr 54.9%

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

   1. pow-sqr 56.1%

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

   2. metadata-eval 56.1%

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

6. Simplified 56.1%

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

7. Taylor expanded in x around 0 47.2%

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

8. Taylor expanded in x around inf 2.3%

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

   1. *-commutative 2.3%

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

10. Simplified 2.3%

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

11. Final simplification 2.3%

    \[\leadsto x \cdot 0.5 + -1 \]
12. 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 2024018 
(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)))))