2isqrt (example 3.6)

Percentage Accurate: 69.1% → 99.8%

Time: 18.3s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sqrt{1 + x}}\\ \mathbf{if}\;\frac{1}{\sqrt{x}} + t_0 \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(0.3125, {x}^{-3.5}, 0.5 \cdot {x}^{-1.5}\right) + -0.375 \cdot {x}^{-2.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sqrt (+ 1.0 x)))))
   (if (<= (+ (/ 1.0 (sqrt x)) t_0) 2e-10)
     (+ (fma 0.3125 (pow x -3.5) (* 0.5 (pow x -1.5))) (* -0.375 (pow x -2.5)))
     (+ (pow x -0.5) t_0))))

C

double code(double x) {
	double t_0 = -1.0 / sqrt((1.0 + x));
	double tmp;
	if (((1.0 / sqrt(x)) + t_0) <= 2e-10) {
		tmp = fma(0.3125, pow(x, -3.5), (0.5 * pow(x, -1.5))) + (-0.375 * pow(x, -2.5));
	} else {
		tmp = pow(x, -0.5) + t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(-1.0 / sqrt(Float64(1.0 + x)))
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + t_0) <= 2e-10)
		tmp = Float64(fma(0.3125, (x ^ -3.5), Float64(0.5 * (x ^ -1.5))) + Float64(-0.375 * (x ^ -2.5)));
	else
		tmp = Float64((x ^ -0.5) + t_0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], 2e-10], N[(N[(0.3125 * N[Power[x, -3.5], $MachinePrecision] + N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[Power[x, -2.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 2.00000000000000007e-10

   1. Initial program 39.4%

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

      1. reciprocal-define 4.6%

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

      2. reciprocal-define 37.9%

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

      3. +-commutative 37.9%

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

   3. Simplified 37.9%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 25.2%

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

      2. *-un-lft-identity 25.2%

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

      3. *-commutative 25.2%

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

      4. prod-diff 4.8%

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

   6. Applied egg-rr 39.4%

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

      1. associate-+l- 39.4%

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

      2. expm1-log1p 39.4%

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

      3. expm1-def 5.6%

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

      4. associate--l- 5.6%

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

      5. fma-udef 5.6%

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

      6. distribute-lft1-in 5.6%

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

      7. metadata-eval 5.6%

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

      8. mul0-lft 5.6%

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

      9. metadata-eval 5.6%

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

      10. expm1-def 39.4%

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

      11. expm1-log1p 39.4%

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

      12. +-commutative 39.4%

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

   8. Simplified 39.4%

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

   9. Taylor expanded in x around inf 68.6%

      \[\leadsto \color{blue}{-0.375 \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(0.3125 \cdot \sqrt{\frac{1}{{x}^{7}}} + 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)} \]
   10. Step-by-step derivation

       1. +-commutative 68.6%

          \[\leadsto \color{blue}{\left(0.3125 \cdot \sqrt{\frac{1}{{x}^{7}}} + 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right) + -0.375 \cdot \sqrt{\frac{1}{{x}^{5}}}} \]

       2. *-un-lft-identity 68.6%

          \[\leadsto \color{blue}{1 \cdot \left(0.3125 \cdot \sqrt{\frac{1}{{x}^{7}}} + 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\right)} + -0.375 \cdot \sqrt{\frac{1}{{x}^{5}}} \]

       3. fma-def 68.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(1, 0.3125 \cdot \sqrt{\frac{1}{{x}^{7}}} + 0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}, -0.375 \cdot \sqrt{\frac{1}{{x}^{5}}}\right)} \]

   11. Applied egg-rr 99.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(1, \mathsf{fma}\left(0.3125, {x}^{-3.5}, {x}^{-1.5} \cdot 0.5\right), -0.375 \cdot {x}^{-2.5}\right)} \]
   12. Step-by-step derivation

       1. fma-udef 99.9%

          \[\leadsto \color{blue}{1 \cdot \mathsf{fma}\left(0.3125, {x}^{-3.5}, {x}^{-1.5} \cdot 0.5\right) + -0.375 \cdot {x}^{-2.5}} \]

       2. *-lft-identity 99.9%

          \[\leadsto \color{blue}{\mathsf{fma}\left(0.3125, {x}^{-3.5}, {x}^{-1.5} \cdot 0.5\right)} + -0.375 \cdot {x}^{-2.5} \]

       3. *-commutative 99.9%

          \[\leadsto \mathsf{fma}\left(0.3125, {x}^{-3.5}, \color{blue}{0.5 \cdot {x}^{-1.5}}\right) + -0.375 \cdot {x}^{-2.5} \]

   13. Simplified 99.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.3125, {x}^{-3.5}, 0.5 \cdot {x}^{-1.5}\right) + -0.375 \cdot {x}^{-2.5}} \]

   if 2.00000000000000007e-10 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.6%

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

      1. reciprocal-define 39.3%

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

      2. reciprocal-define 39.3%

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

      3. +-commutative 39.3%

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

   3. Simplified 39.3%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.3%

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

      2. *-un-lft-identity 39.3%

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

      3. *-commutative 39.3%

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

      4. prod-diff 39.3%

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

   6. Applied egg-rr 99.9%

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

      1. associate-+l- 99.9%

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

      2. expm1-log1p 99.9%

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

      3. expm1-def 99.9%

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

      4. associate--l- 99.9%

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

      5. fma-udef 99.9%

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

      6. distribute-lft1-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. mul0-lft 99.9%

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

      9. metadata-eval 99.9%

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

      10. expm1-def 99.9%

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

      11. expm1-log1p 99.9%

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

      12. +-commutative 99.9%

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

   8. Simplified 99.9%

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

      1. metadata-eval 99.9%

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

      2. pow-flip 99.9%

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

      3. pow1/2 99.9%

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

      4. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(0.3125, {x}^{-3.5}, 0.5 \cdot {x}^{-1.5}\right) + -0.375 \cdot {x}^{-2.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} + \frac{-1}{\sqrt{1 + x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 64000000:\\ \;\;\;\;{x}^{-0.5} - \frac{\sqrt[3]{{\left(1 + x\right)}^{-0.5}}}{\sqrt[3]{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 64000000.0)
   (- (pow x -0.5) (/ (cbrt (pow (+ 1.0 x) -0.5)) (cbrt (+ 1.0 x))))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 64000000.0) {
		tmp = pow(x, -0.5) - (cbrt(pow((1.0 + x), -0.5)) / cbrt((1.0 + x)));
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 64000000.0) {
		tmp = Math.pow(x, -0.5) - (Math.cbrt(Math.pow((1.0 + x), -0.5)) / Math.cbrt((1.0 + x)));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 64000000.0)
		tmp = Float64((x ^ -0.5) - Float64(cbrt((Float64(1.0 + x) ^ -0.5)) / cbrt(Float64(1.0 + x))));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 64000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[(N[Power[N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision], 1/3], $MachinePrecision] / N[Power[N[(1.0 + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.4e7

   1. Initial program 99.4%

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

      1. reciprocal-define 39.1%

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

      2. reciprocal-define 39.0%

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

      3. +-commutative 39.0%

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

   3. Simplified 39.0%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.1%

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

      2. *-un-lft-identity 39.1%

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

      3. *-commutative 39.1%

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

      4. prod-diff 39.1%

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

   6. Applied egg-rr 99.8%

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

      1. associate-+l- 99.8%

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

      2. expm1-log1p 99.8%

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

      3. expm1-def 99.5%

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

      4. associate--l- 99.5%

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

      5. fma-udef 99.5%

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

      6. distribute-lft1-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. mul0-lft 99.5%

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

      9. metadata-eval 99.5%

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

      10. expm1-def 99.8%

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

      11. expm1-log1p 99.8%

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

      12. +-commutative 99.8%

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

   8. Simplified 99.8%

      \[\leadsto \color{blue}{{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}} \]
   9. 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 99.7%

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

      3. metadata-eval 99.7%

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

      4. pow1/2 99.7%

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

      5. add-cbrt-cube 99.6%

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

      6. add-sqr-sqrt 99.6%

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

      7. metadata-eval 99.6%

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

      8. add-sqr-sqrt 99.6%

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

      9. hypot-udef 99.6%

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

      10. cbrt-prod 99.6%

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

      11. hypot-udef 99.6%

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

      12. metadata-eval 99.6%

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

      13. add-sqr-sqrt 99.6%

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

      14. frac-times 99.6%

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

      15. metadata-eval 99.6%

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

      16. metadata-eval 99.6%

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

      17. add-sqr-sqrt 99.6%

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

   10. Applied egg-rr 99.8%

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

   if 6.4e7 < x

   1. Initial program 39.1%

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

      1. reciprocal-define 4.5%

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

      2. reciprocal-define 38.2%

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

      3. +-commutative 38.2%

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

   3. Simplified 38.2%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 25.3%

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

      2. *-un-lft-identity 25.3%

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

      3. *-commutative 25.3%

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

      4. prod-diff 4.7%

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

   6. Applied egg-rr 39.1%

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

      1. associate-+l- 39.1%

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

      2. expm1-log1p 39.1%

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

      3. expm1-def 5.2%

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

      4. associate--l- 5.2%

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

      5. fma-udef 5.2%

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

      6. distribute-lft1-in 5.2%

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

      7. metadata-eval 5.2%

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

      8. mul0-lft 5.2%

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

      9. metadata-eval 5.2%

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

      10. expm1-def 39.1%

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

      11. expm1-log1p 39.1%

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

      12. +-commutative 39.1%

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

   8. Simplified 39.1%

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

   9. Taylor expanded in x around inf 68.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 64000000:\\ \;\;\;\;{x}^{-0.5} - \frac{\sqrt[3]{{\left(1 + x\right)}^{-0.5}}}{\sqrt[3]{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 300000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 300000000.0)
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 300000000.0) {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 300000000.0d0) then
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 300000000.0) {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 300000000.0:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	else:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 300000000.0)
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 300000000.0)
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	else
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 300000000.0], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3e8

   1. Initial program 99.4%

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

      1. reciprocal-define 39.1%

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

      2. reciprocal-define 39.0%

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

      3. +-commutative 39.0%

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

   3. Simplified 39.0%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.1%

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

      2. *-un-lft-identity 39.1%

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

      3. *-commutative 39.1%

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

      4. prod-diff 39.1%

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

   6. Applied egg-rr 99.8%

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

      1. associate-+l- 99.8%

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

      2. expm1-log1p 99.8%

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

      3. expm1-def 99.5%

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

      4. associate--l- 99.5%

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

      5. fma-udef 99.5%

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

      6. distribute-lft1-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. mul0-lft 99.5%

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

      9. metadata-eval 99.5%

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

      10. expm1-def 99.8%

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

      11. expm1-log1p 99.8%

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

      12. +-commutative 99.8%

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

   8. Simplified 99.8%

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

   if 3e8 < x

   1. Initial program 39.1%

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

      1. reciprocal-define 4.5%

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

      2. reciprocal-define 38.2%

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

      3. +-commutative 38.2%

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

   3. Simplified 38.2%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 25.3%

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

      2. *-un-lft-identity 25.3%

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

      3. *-commutative 25.3%

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

      4. prod-diff 4.7%

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

   6. Applied egg-rr 39.1%

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

      1. associate-+l- 39.1%

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

      2. expm1-log1p 39.1%

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

      3. expm1-def 5.2%

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

      4. associate--l- 5.2%

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

      5. fma-udef 5.2%

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

      6. distribute-lft1-in 5.2%

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

      7. metadata-eval 5.2%

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

      8. mul0-lft 5.2%

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

      9. metadata-eval 5.2%

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

      10. expm1-def 39.1%

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

      11. expm1-log1p 39.1%

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

      12. +-commutative 39.1%

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

   8. Simplified 39.1%

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

   9. Taylor expanded in x around inf 68.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 300000000:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7)
   (- (pow x -0.5) (/ 1.0 (+ 1.0 (* x 0.5))))
   (* 0.5 (sqrt (/ 1.0 (pow x 3.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * sqrt((1.0 / pow(x, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.7d0) then
        tmp = (x ** (-0.5d0)) - (1.0d0 / (1.0d0 + (x * 0.5d0)))
    else
        tmp = 0.5d0 * sqrt((1.0d0 / (x ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = Math.pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = 0.5 * Math.sqrt((1.0 / Math.pow(x, 3.0)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = math.pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = 0.5 * math.sqrt((1.0 / math.pow(x, 3.0)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = Float64((x ^ -0.5) - Float64(1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = Float64(0.5 * sqrt(Float64(1.0 / (x ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = (x ^ -0.5) - (1.0 / (1.0 + (x * 0.5)));
	else
		tmp = 0.5 * sqrt((1.0 / (x ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[Sqrt[N[(1.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.6%

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

      1. reciprocal-define 39.4%

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

      2. reciprocal-define 39.5%

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

      3. +-commutative 39.5%

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

   3. Simplified 39.5%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.5%

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

      2. *-un-lft-identity 39.5%

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

      3. *-commutative 39.5%

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

      4. prod-diff 39.5%

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

   6. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. mul0-lft 100.0%

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

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

      12. +-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. metadata-eval 100.0%

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

      2. pow-flip 100.0%

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

      3. pow1/2 100.0%

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

      4. +-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in x around 0 98.9%

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

       1. *-commutative 98.9%

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

   13. Simplified 98.9%

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

   if 1.69999999999999996 < x

   1. Initial program 40.3%

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

      1. reciprocal-define 5.0%

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

      2. reciprocal-define 37.8%

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

      3. +-commutative 37.8%

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

   3. Simplified 37.8%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 25.3%

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

      2. *-un-lft-identity 25.3%

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

      3. *-commutative 25.3%

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

      4. prod-diff 5.2%

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

   6. Applied egg-rr 40.3%

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

      1. associate-+l- 40.3%

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

      2. expm1-log1p 40.3%

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

      3. expm1-def 6.9%

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

      4. associate--l- 6.9%

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

      5. fma-udef 6.9%

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

      6. distribute-lft1-in 6.9%

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

      7. metadata-eval 6.9%

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

      8. mul0-lft 6.9%

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

      9. metadata-eval 6.9%

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

      10. expm1-def 40.3%

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

      11. expm1-log1p 40.3%

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

      12. +-commutative 40.3%

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

   8. Simplified 40.3%

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

   9. Taylor expanded in x around inf 67.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\frac{1}{{x}^{3}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.4:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{reciprocal}\left(\left({x}^{3}\right)\right)} \cdot 1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.4)
   (- (pow x -0.5) (/ 1.0 (+ 1.0 (* x 0.5))))
   (* (sqrt (reciprocal (pow x 3.0))) 1.5)))

Derivation

1. Split input into 2 regimes

2. if x < 4.4000000000000004

   1. Initial program 99.6%

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

      1. reciprocal-define 39.4%

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

      2. reciprocal-define 39.4%

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

      3. +-commutative 39.4%

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

   3. Simplified 39.4%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.4%

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

      2. *-un-lft-identity 39.4%

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

      3. *-commutative 39.4%

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

      4. prod-diff 39.4%

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

   6. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 99.9%

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

      4. associate--l- 99.9%

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

      5. fma-udef 99.9%

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

      6. distribute-lft1-in 99.9%

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

      7. metadata-eval 99.9%

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

      8. mul0-lft 99.9%

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

      9. metadata-eval 99.9%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

      12. +-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. metadata-eval 100.0%

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

      2. pow-flip 100.0%

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

      3. pow1/2 100.0%

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

      4. +-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in x around 0 98.3%

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

       1. *-commutative 98.3%

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

   13. Simplified 98.3%

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

   if 4.4000000000000004 < x

   1. Initial program 39.9%

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

      1. reciprocal-define 4.8%

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

      2. reciprocal-define 37.8%

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

      3. +-commutative 37.8%

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

   3. Simplified 37.8%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 25.2%

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

      2. *-un-lft-identity 25.2%

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

      3. *-commutative 25.2%

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

      4. prod-diff 5.0%

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

   6. Applied egg-rr 39.9%

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

      1. associate-+l- 39.9%

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

      2. expm1-log1p 39.9%

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

      3. expm1-def 6.2%

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

      4. associate--l- 6.2%

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

      5. fma-udef 6.2%

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

      6. distribute-lft1-in 6.2%

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

      7. metadata-eval 6.2%

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

      8. mul0-lft 6.2%

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

      9. metadata-eval 6.2%

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

      10. expm1-def 39.9%

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

      11. expm1-log1p 39.9%

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

      12. +-commutative 39.9%

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

   8. Simplified 39.9%

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

   10. Applied egg-rr 39.8%

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

   11. Taylor expanded in x around inf 42.2%

       \[\leadsto \color{blue}{1.5 \cdot \sqrt{\frac{1}{{x}^{3}}}} \]
   12. Step-by-step derivation

       1. reciprocal-define 42.2%

          \[\leadsto 1.5 \cdot \sqrt{\color{blue}{\mathsf{reciprocal}\left(\left({x}^{3}\right)\right)}} \]

   13. Simplified 42.2%

       \[\leadsto \color{blue}{1.5 \cdot \sqrt{\mathsf{reciprocal}\left(\left({x}^{3}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.4:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{reciprocal}\left(\left({x}^{3}\right)\right)} \cdot 1.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{reciprocal}\left(\left({x}^{3}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7)
   (- (pow x -0.5) (/ 1.0 (+ 1.0 (* x 0.5))))
   (* 0.5 (sqrt (reciprocal (pow x 3.0))))))

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.6%

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

      1. reciprocal-define 39.4%

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

      2. reciprocal-define 39.5%

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

      3. +-commutative 39.5%

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

   3. Simplified 39.5%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 39.5%

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

      2. *-un-lft-identity 39.5%

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

      3. *-commutative 39.5%

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

      4. prod-diff 39.5%

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

   6. Applied egg-rr 100.0%

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

      1. associate-+l- 100.0%

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

      2. expm1-log1p 100.0%

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

      3. expm1-def 100.0%

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

      4. associate--l- 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-lft1-in 100.0%

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

      7. metadata-eval 100.0%

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

      8. mul0-lft 100.0%

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

      9. metadata-eval 100.0%

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

      10. expm1-def 100.0%

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

      11. expm1-log1p 100.0%

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

      12. +-commutative 100.0%

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

   8. Simplified 100.0%

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

      1. metadata-eval 100.0%

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

      2. pow-flip 100.0%

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

      3. pow1/2 100.0%

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

      4. +-commutative 100.0%

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

   10. Applied egg-rr 100.0%

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

   11. Taylor expanded in x around 0 98.9%

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

       1. *-commutative 98.9%

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

   13. Simplified 98.9%

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

   if 1.69999999999999996 < x

   1. Initial program 40.3%

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

      1. reciprocal-define 5.0%

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

      2. reciprocal-define 37.8%

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

      3. +-commutative 37.8%

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

   3. Simplified 37.8%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 25.3%

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

      2. *-un-lft-identity 25.3%

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

      3. *-commutative 25.3%

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

      4. prod-diff 5.2%

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

   6. Applied egg-rr 40.3%

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

      1. associate-+l- 40.3%

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

      2. expm1-log1p 40.3%

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

      3. expm1-def 6.9%

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

      4. associate--l- 6.9%

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

      5. fma-udef 6.9%

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

      6. distribute-lft1-in 6.9%

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

      7. metadata-eval 6.9%

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

      8. mul0-lft 6.9%

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

      9. metadata-eval 6.9%

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

      10. expm1-def 40.3%

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

      11. expm1-log1p 40.3%

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

      12. +-commutative 40.3%

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

   8. Simplified 40.3%

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

      1. add-exp-log 6.7%

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

      2. log-pow 6.7%

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

      3. log1p-udef 6.7%

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

   10. Applied egg-rr 6.7%

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

   12. Simplified 6.6%

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

   13. Taylor expanded in x around inf 67.2%

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

       1. *-commutative 67.2%

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

       2. reciprocal-define 49.6%

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

   15. Simplified 49.6%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{\mathsf{reciprocal}\left(\left({x}^{3}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+23}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left({x}^{2}\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2e+23)
   (- (pow x -0.5) (/ 1.0 (+ 1.0 (* x 0.5))))
   (pow (pow x 2.0) -0.25)))

C

double code(double x) {
	double tmp;
	if (x <= 2.2e+23) {
		tmp = pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = pow(pow(x, 2.0), -0.25);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.2d+23) then
        tmp = (x ** (-0.5d0)) - (1.0d0 / (1.0d0 + (x * 0.5d0)))
    else
        tmp = (x ** 2.0d0) ** (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2e+23) {
		tmp = Math.pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = Math.pow(Math.pow(x, 2.0), -0.25);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2e+23:
		tmp = math.pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)))
	else:
		tmp = math.pow(math.pow(x, 2.0), -0.25)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2e+23)
		tmp = Float64((x ^ -0.5) - Float64(1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = (x ^ 2.0) ^ -0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2e+23)
		tmp = (x ^ -0.5) - (1.0 / (1.0 + (x * 0.5)));
	else
		tmp = (x ^ 2.0) ^ -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2e+23], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[x, 2.0], $MachinePrecision], -0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000008e23

   1. Initial program 96.7%

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

      1. reciprocal-define 37.9%

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

      2. reciprocal-define 37.7%

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

      3. +-commutative 37.7%

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

   3. Simplified 37.7%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 37.8%

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

      2. *-un-lft-identity 37.8%

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

      3. *-commutative 37.8%

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

      4. prod-diff 38.0%

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

   6. Applied egg-rr 97.0%

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

      1. associate-+l- 97.0%

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

      2. expm1-log1p 97.0%

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

      3. expm1-def 96.0%

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

      4. associate--l- 96.0%

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

      5. fma-udef 96.0%

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

      6. distribute-lft1-in 96.0%

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

      7. metadata-eval 96.0%

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

      8. mul0-lft 96.0%

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

      9. metadata-eval 96.0%

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

      10. expm1-def 97.0%

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

      11. expm1-log1p 97.0%

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

      12. +-commutative 97.0%

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

   8. Simplified 97.0%

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

      1. metadata-eval 97.0%

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

      2. pow-flip 96.9%

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

      3. pow1/2 96.9%

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

      4. +-commutative 96.9%

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

   10. Applied egg-rr 96.9%

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

   11. Taylor expanded in x around 0 93.6%

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

       1. *-commutative 93.6%

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

   13. Simplified 93.6%

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

   if 2.20000000000000008e23 < x

   1. Initial program 39.6%

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

      1. reciprocal-define 4.4%

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

      2. reciprocal-define 39.6%

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

      3. +-commutative 39.6%

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

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 26.1%

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

      2. *-un-lft-identity 26.1%

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

      3. *-commutative 26.1%

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

      4. prod-diff 4.5%

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

   6. Applied egg-rr 39.6%

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

      1. associate-+l- 39.6%

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

      2. expm1-log1p 39.6%

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

      3. expm1-def 5.1%

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

      4. associate--l- 5.1%

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

      5. fma-udef 5.1%

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

      6. distribute-lft1-in 5.1%

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

      7. metadata-eval 5.1%

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

      8. mul0-lft 5.1%

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

      9. metadata-eval 5.1%

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

      10. expm1-def 39.6%

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

      11. expm1-log1p 39.6%

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

      12. +-commutative 39.6%

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

   8. Simplified 39.6%

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

      1. metadata-eval 39.6%

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

      2. pow-flip 32.1%

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

      3. pow1/2 32.1%

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

      4. +-commutative 32.1%

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

   10. Applied egg-rr 32.1%

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

   11. Taylor expanded in x around inf 5.2%

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

       1. reciprocal-define 5.2%

          \[\leadsto \sqrt{\color{blue}{\mathsf{reciprocal}\left(x\right)}} \]

   13. Simplified 5.2%

       \[\leadsto \color{blue}{\sqrt{\mathsf{reciprocal}\left(x\right)}} \]
   14. Step-by-step derivation

       1. reciprocal-define 5.2%

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

       2. pow1/2 5.2%

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

       3. inv-pow 5.2%

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

       4. pow-pow 5.2%

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

       5. metadata-eval 5.2%

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

       6. sqr-pow 5.2%

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

       7. pow-prod-down 40.4%

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

       8. pow2 40.4%

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

       9. metadata-eval 40.4%

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

   15. Applied egg-rr 40.4%

       \[\leadsto \color{blue}{{\left({x}^{2}\right)}^{-0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+23}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left({x}^{2}\right)}^{-0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+23}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{-1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2e+23)
   (- (pow x -0.5) (/ 1.0 (+ 1.0 (* x 0.5))))
   (cbrt (pow x -1.5))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2e+23) {
		tmp = pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = cbrt(pow(x, -1.5));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2e+23) {
		tmp = Math.pow(x, -0.5) - (1.0 / (1.0 + (x * 0.5)));
	} else {
		tmp = Math.cbrt(Math.pow(x, -1.5));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2e+23)
		tmp = Float64((x ^ -0.5) - Float64(1.0 / Float64(1.0 + Float64(x * 0.5))));
	else
		tmp = cbrt((x ^ -1.5));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2e+23], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 / N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[Power[x, -1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.20000000000000008e23

   1. Initial program 96.7%

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

      1. reciprocal-define 37.9%

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

      2. reciprocal-define 37.7%

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

      3. +-commutative 37.7%

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

   3. Simplified 37.7%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 37.8%

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

      2. *-un-lft-identity 37.8%

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

      3. *-commutative 37.8%

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

      4. prod-diff 38.0%

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

   6. Applied egg-rr 97.0%

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

      1. associate-+l- 97.0%

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

      2. expm1-log1p 97.0%

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

      3. expm1-def 96.0%

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

      4. associate--l- 96.0%

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

      5. fma-udef 96.0%

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

      6. distribute-lft1-in 96.0%

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

      7. metadata-eval 96.0%

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

      8. mul0-lft 96.0%

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

      9. metadata-eval 96.0%

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

      10. expm1-def 97.0%

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

      11. expm1-log1p 97.0%

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

      12. +-commutative 97.0%

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

   8. Simplified 97.0%

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

      1. metadata-eval 97.0%

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

      2. pow-flip 96.9%

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

      3. pow1/2 96.9%

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

      4. +-commutative 96.9%

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

   10. Applied egg-rr 96.9%

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

   11. Taylor expanded in x around 0 93.6%

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

       1. *-commutative 93.6%

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

   13. Simplified 93.6%

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

   if 2.20000000000000008e23 < x

   1. Initial program 39.6%

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

      1. reciprocal-define 4.4%

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

      2. reciprocal-define 39.6%

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

      3. +-commutative 39.6%

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

   3. Simplified 39.6%

      \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 26.1%

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

      2. *-un-lft-identity 26.1%

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

      3. *-commutative 26.1%

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

      4. prod-diff 4.5%

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

   6. Applied egg-rr 39.6%

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

      1. associate-+l- 39.6%

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

      2. expm1-log1p 39.6%

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

      3. expm1-def 5.1%

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

      4. associate--l- 5.1%

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

      5. fma-udef 5.1%

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

      6. distribute-lft1-in 5.1%

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

      7. metadata-eval 5.1%

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

      8. mul0-lft 5.1%

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

      9. metadata-eval 5.1%

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

      10. expm1-def 39.6%

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

      11. expm1-log1p 39.6%

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

      12. +-commutative 39.6%

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

   8. Simplified 39.6%

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

      1. metadata-eval 39.6%

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

      2. pow-flip 32.1%

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

      3. pow1/2 32.1%

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

      4. +-commutative 32.1%

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

   10. Applied egg-rr 32.1%

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

   11. Taylor expanded in x around inf 5.2%

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

       1. reciprocal-define 5.2%

          \[\leadsto \sqrt{\color{blue}{\mathsf{reciprocal}\left(x\right)}} \]

   13. Simplified 5.2%

       \[\leadsto \color{blue}{\sqrt{\mathsf{reciprocal}\left(x\right)}} \]
   14. Step-by-step derivation

       1. reciprocal-define 5.2%

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

       2. add-cbrt-cube 36.6%

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

       3. pow3 36.6%

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

       4. reciprocal-define 36.6%

          \[\leadsto \sqrt[3]{{\left(\sqrt{\color{blue}{\mathsf{reciprocal}\left(x\right)}}\right)}^{3}} \]

       5. reciprocal-define 36.6%

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

       6. pow1/2 36.6%

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

       7. inv-pow 36.6%

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

       8. pow-pow 36.6%

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

       9. metadata-eval 36.6%

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

       10. pow-pow 36.6%

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

       11. metadata-eval 36.6%

           \[\leadsto \sqrt[3]{{x}^{\color{blue}{-1.5}}} \]

       12. pow1/3 36.6%

           \[\leadsto \color{blue}{{\left({x}^{-1.5}\right)}^{0.3333333333333333}} \]

   15. Applied egg-rr 36.6%

       \[\leadsto \color{blue}{{\left({x}^{-1.5}\right)}^{0.3333333333333333}} \]
   16. Step-by-step derivation

       1. unpow1/3 36.6%

          \[\leadsto \color{blue}{\sqrt[3]{{x}^{-1.5}}} \]

   17. Simplified 36.6%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+23}:\\ \;\;\;\;{x}^{-0.5} - \frac{1}{1 + x \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{x}^{-1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.8%

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

   1. reciprocal-define 21.5%

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

   2. reciprocal-define 38.6%

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

   3. +-commutative 38.6%

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

3. Simplified 38.6%

   \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 32.1%

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

   2. *-un-lft-identity 32.1%

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

   3. *-commutative 32.1%

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

   4. prod-diff 21.6%

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

6. Applied egg-rr 69.0%

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

   1. associate-+l- 69.0%

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

   2. expm1-log1p 69.0%

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

   3. expm1-def 51.6%

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

   4. associate--l- 51.6%

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

   5. fma-udef 51.6%

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

   6. distribute-lft1-in 51.6%

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

   7. metadata-eval 51.6%

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

   8. mul0-lft 51.6%

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

   9. metadata-eval 51.6%

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

   10. expm1-def 69.0%

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

   11. expm1-log1p 69.0%

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

   12. +-commutative 69.0%

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

8. Simplified 69.0%

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

   1. metadata-eval 69.0%

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

   2. pow-flip 65.3%

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

   3. pow1/2 65.3%

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

   4. +-commutative 65.3%

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

10. Applied egg-rr 65.3%

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

11. Taylor expanded in x around 0 50.5%

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

    1. *-commutative 50.5%

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

13. Simplified 50.5%

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

14. Final simplification 50.5%

    \[\leadsto {x}^{-0.5} - \frac{1}{1 + x \cdot 0.5} \]
15. Add Preprocessing

Alternative 10: 51.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.8%

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

   1. reciprocal-define 21.5%

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

   2. reciprocal-define 38.6%

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

   3. +-commutative 38.6%

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

3. Simplified 38.6%

   \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 32.1%

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

   2. *-un-lft-identity 32.1%

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

   3. *-commutative 32.1%

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

   4. prod-diff 21.6%

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

6. Applied egg-rr 69.0%

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

   1. associate-+l- 69.0%

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

   2. expm1-log1p 69.0%

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

   3. expm1-def 51.6%

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

   4. associate--l- 51.6%

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

   5. fma-udef 51.6%

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

   6. distribute-lft1-in 51.6%

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

   7. metadata-eval 51.6%

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

   8. mul0-lft 51.6%

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

   9. metadata-eval 51.6%

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

   10. expm1-def 69.0%

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

   11. expm1-log1p 69.0%

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

   12. +-commutative 69.0%

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

8. Simplified 69.0%

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

9. Taylor expanded in x around 0 49.3%

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

10. Final simplification 49.3%

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

Alternative 11: 51.0% 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 68.8%

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

   1. reciprocal-define 21.5%

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

   2. reciprocal-define 38.6%

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

   3. +-commutative 38.6%

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

3. Simplified 38.6%

   \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 32.1%

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

   2. *-un-lft-identity 32.1%

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

   3. *-commutative 32.1%

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

   4. prod-diff 21.6%

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

6. Applied egg-rr 69.0%

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

   1. associate-+l- 69.0%

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

   2. expm1-log1p 69.0%

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

   3. expm1-def 51.6%

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

   4. associate--l- 51.6%

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

   5. fma-udef 51.6%

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

   6. distribute-lft1-in 51.6%

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

   7. metadata-eval 51.6%

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

   8. mul0-lft 51.6%

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

   9. metadata-eval 51.6%

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

   10. expm1-def 69.0%

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

   11. expm1-log1p 69.0%

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

   12. +-commutative 69.0%

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

8. Simplified 69.0%

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

   1. metadata-eval 69.0%

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

   2. pow-flip 65.3%

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

   3. pow1/2 65.3%

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

   4. +-commutative 65.3%

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

10. Applied egg-rr 65.3%

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

11. Taylor expanded in x around inf 48.5%

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

    1. reciprocal-define 22.6%

       \[\leadsto \sqrt{\color{blue}{\mathsf{reciprocal}\left(x\right)}} \]

    2. unpow1/2 22.6%

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

    3. rem-exp-log 22.6%

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

    4. reciprocal-define 45.2%

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

    5. exp-neg 45.2%

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

    6. exp-prod 45.2%

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

    7. distribute-lft-neg-out 45.2%

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

    8. distribute-rgt-neg-in 45.2%

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

    9. metadata-eval 45.2%

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

    10. exp-to-pow 48.5%

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

13. Simplified 48.5%

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

14. Final simplification 48.5%

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

Alternative 12: 22.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (reciprocal (sqrt x)))

Derivation

1. Initial program 68.8%

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

   1. reciprocal-define 21.5%

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

   2. reciprocal-define 38.6%

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

   3. +-commutative 38.6%

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

3. Simplified 38.6%

   \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 32.1%

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

   2. *-un-lft-identity 32.1%

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

   3. *-commutative 32.1%

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

   4. prod-diff 21.6%

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

6. Applied egg-rr 69.0%

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

   1. associate-+l- 69.0%

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

   2. expm1-log1p 69.0%

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

   3. expm1-def 51.6%

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

   4. associate--l- 51.6%

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

   5. fma-udef 51.6%

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

   6. distribute-lft1-in 51.6%

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

   7. metadata-eval 51.6%

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

   8. mul0-lft 51.6%

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

   9. metadata-eval 51.6%

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

   10. expm1-def 69.0%

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

   11. expm1-log1p 69.0%

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

   12. +-commutative 69.0%

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

8. Simplified 69.0%

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

   1. metadata-eval 69.0%

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

   2. pow-flip 65.3%

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

   3. pow1/2 65.3%

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

   4. +-commutative 65.3%

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

10. Applied egg-rr 65.3%

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

11. Taylor expanded in x around inf 48.5%

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

    1. reciprocal-define 22.6%

       \[\leadsto \sqrt{\color{blue}{\mathsf{reciprocal}\left(x\right)}} \]

13. Simplified 22.6%

    \[\leadsto \color{blue}{\sqrt{\mathsf{reciprocal}\left(x\right)}} \]
14. Step-by-step derivation

    1. reciprocal-define 48.5%

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

    2. pow1/2 48.5%

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

    3. inv-pow 48.5%

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

    4. pow-pow 48.5%

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

    5. metadata-eval 48.5%

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

    6. add-sqr-sqrt 48.4%

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

    7. unpow-prod-down 48.1%

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

15. Applied egg-rr 48.1%

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

    1. pow-sqr 48.4%

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

    2. metadata-eval 48.4%

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

    3. unpow-1 48.4%

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

    4. reciprocal-define 21.6%

       \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right)} \]

17. Simplified 21.6%

    \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right)} \]

18. Final simplification 21.6%

    \[\leadsto \mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) \]
19. Add Preprocessing

Alternative 13: 23.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{reciprocal}\left(x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (reciprocal x)))

Derivation

1. Initial program 68.8%

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

   1. reciprocal-define 21.5%

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

   2. reciprocal-define 38.6%

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

   3. +-commutative 38.6%

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

3. Simplified 38.6%

   \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 32.1%

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

   2. *-un-lft-identity 32.1%

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

   3. *-commutative 32.1%

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

   4. prod-diff 21.6%

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

6. Applied egg-rr 69.0%

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

   1. associate-+l- 69.0%

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

   2. expm1-log1p 69.0%

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

   3. expm1-def 51.6%

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

   4. associate--l- 51.6%

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

   5. fma-udef 51.6%

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

   6. distribute-lft1-in 51.6%

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

   7. metadata-eval 51.6%

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

   8. mul0-lft 51.6%

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

   9. metadata-eval 51.6%

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

   10. expm1-def 69.0%

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

   11. expm1-log1p 69.0%

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

   12. +-commutative 69.0%

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

8. Simplified 69.0%

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

   1. metadata-eval 69.0%

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

   2. pow-flip 65.3%

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

   3. pow1/2 65.3%

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

   4. +-commutative 65.3%

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

10. Applied egg-rr 65.3%

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

11. Taylor expanded in x around inf 48.5%

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

    1. reciprocal-define 22.6%

       \[\leadsto \sqrt{\color{blue}{\mathsf{reciprocal}\left(x\right)}} \]

13. Simplified 22.6%

    \[\leadsto \color{blue}{\sqrt{\mathsf{reciprocal}\left(x\right)}} \]

14. Final simplification 22.6%

    \[\leadsto \sqrt{\mathsf{reciprocal}\left(x\right)} \]
15. Add Preprocessing

Alternative 14: 1.9% accurate, 209.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 68.8%

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

   1. reciprocal-define 21.5%

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

   2. reciprocal-define 38.6%

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

   3. +-commutative 38.6%

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

3. Simplified 38.6%

   \[\leadsto \color{blue}{\mathsf{reciprocal}\left(\left(\sqrt{x}\right)\right) - \mathsf{reciprocal}\left(\left(\sqrt{1 + x}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 20.2%

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

6. Taylor expanded in x around inf 2.0%

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

7. Final simplification 2.0%

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

Developer target: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024024 
(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)))))