2isqrt (example 3.6)

Percentage Accurate: 69.0% → 99.5%

Time: 9.8s

Alternatives: 9

Speedup: 2.0×

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: 69.0% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ (/ 1.0 (sqrt x)) (/ -1.0 (sqrt (+ 1.0 x)))) 5e-16)
   (* 0.5 (pow x -1.5))
   (- (pow x -0.5) (pow (+ 1.0 x) -0.5))))

C

double code(double x) {
	double tmp;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 5e-16) {
		tmp = 0.5 * pow(x, -1.5);
	} else {
		tmp = pow(x, -0.5) - pow((1.0 + x), -0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (((1.0d0 / sqrt(x)) + ((-1.0d0) / sqrt((1.0d0 + x)))) <= 5d-16) then
        tmp = 0.5d0 * (x ** (-1.5d0))
    else
        tmp = (x ** (-0.5d0)) - ((1.0d0 + x) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (((1.0 / Math.sqrt(x)) + (-1.0 / Math.sqrt((1.0 + x)))) <= 5e-16) {
		tmp = 0.5 * Math.pow(x, -1.5);
	} else {
		tmp = Math.pow(x, -0.5) - Math.pow((1.0 + x), -0.5);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if ((1.0 / math.sqrt(x)) + (-1.0 / math.sqrt((1.0 + x)))) <= 5e-16:
		tmp = 0.5 * math.pow(x, -1.5)
	else:
		tmp = math.pow(x, -0.5) - math.pow((1.0 + x), -0.5)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(Float64(1.0 / sqrt(x)) + Float64(-1.0 / sqrt(Float64(1.0 + x)))) <= 5e-16)
		tmp = Float64(0.5 * (x ^ -1.5));
	else
		tmp = Float64((x ^ -0.5) - (Float64(1.0 + x) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (((1.0 / sqrt(x)) + (-1.0 / sqrt((1.0 + x)))) <= 5e-16)
		tmp = 0.5 * (x ^ -1.5);
	else
		tmp = (x ^ -0.5) - ((1.0 + x) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sqrt[N[(1.0 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-16], N[(0.5 * N[Power[x, -1.5], $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] - N[Power[N[(1.0 + x), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1)))) < 5.0000000000000004e-16

   1. Initial program 34.4%

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

      1. frac-sub 34.5%

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

      2. div-inv 34.5%

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

      3. *-un-lft-identity 34.5%

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

      4. +-commutative 34.5%

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

      5. *-rgt-identity 34.5%

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

      6. metadata-eval 34.5%

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

      7. frac-times 34.5%

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

      8. un-div-inv 34.5%

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

      9. pow1/2 34.5%

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

      10. pow-flip 34.5%

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

      11. metadata-eval 34.5%

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

      12. +-commutative 34.5%

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

   3. Applied egg-rr 34.5%

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

      1. associate-*r/ 34.5%

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

      2. *-rgt-identity 34.5%

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

      3. times-frac 34.5%

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

      4. div-sub 34.6%

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

      5. *-inverses 34.6%

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

      6. /-rgt-identity 34.6%

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

   5. Simplified 34.6%

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

   6. Taylor expanded in x around inf 99.3%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-udef 33.9%

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

      3. div-inv 33.9%

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

      4. associate-*l* 33.9%

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

      5. inv-pow 33.9%

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

      6. metadata-eval 33.9%

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

      7. pow-prod-up 33.9%

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

      8. pow3 33.9%

         \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{{\left({x}^{-0.5}\right)}^{3}}\right)} - 1 \]

      9. pow-pow 33.9%

         \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{{x}^{\left(-0.5 \cdot 3\right)}}\right)} - 1 \]

      10. metadata-eval 33.9%

          \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot {x}^{\color{blue}{-1.5}}\right)} - 1 \]

   8. Applied egg-rr 33.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 99.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)\right)} \]

      2. expm1-log1p 99.7%

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

   10. Simplified 99.7%

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

   if 5.0000000000000004e-16 < (-.f64 (/.f64 1 (sqrt.f64 x)) (/.f64 1 (sqrt.f64 (+.f64 x 1))))

   1. Initial program 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. clear-num 99.4%

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

      3. associate-/r/ 99.4%

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

      4. prod-diff 99.4%

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

      5. *-un-lft-identity 99.4%

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

      6. fma-neg 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. inv-pow 99.4%

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

      9. sqrt-pow2 99.7%

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

      10. metadata-eval 99.7%

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

      11. pow1/2 99.7%

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

      12. pow-flip 99.7%

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

      13. +-commutative 99.7%

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

      14. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. fma-udef 99.7%

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

      2. distribute-lft1-in 99.7%

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

      3. metadata-eval 99.7%

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

      4. mul0-lft 99.7%

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

      5. +-rgt-identity 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{x}} + \frac{-1}{\sqrt{1 + x}} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;0.5 \cdot {x}^{-1.5}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} - {\left(1 + x\right)}^{-0.5}\\ \end{array} \]

Alternative 2: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.7%

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

   1. flip-- 67.6%

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

   2. clear-num 67.5%

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

   3. pow1/2 67.5%

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

   4. pow-flip 67.4%

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

   5. metadata-eval 67.4%

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

   6. inv-pow 67.4%

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

   7. sqrt-pow2 67.4%

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

   8. +-commutative 67.4%

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

   9. metadata-eval 67.4%

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

   10. frac-times 59.6%

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

   11. metadata-eval 59.6%

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

   12. add-sqr-sqrt 60.0%

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

   13. frac-times 61.8%

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

   14. metadata-eval 61.8%

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

   15. add-sqr-sqrt 67.5%

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

3. Applied egg-rr 67.5%

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

   1. frac-sub 68.4%

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

   2. *-un-lft-identity 68.4%

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

5. Applied egg-rr 68.4%

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

   1. *-rgt-identity 68.4%

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

   2. associate--l+ 91.3%

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

   3. +-inverses 91.3%

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

   4. metadata-eval 91.3%

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

7. Simplified 91.3%

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

   1. associate-/r/ 91.3%

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

   2. /-rgt-identity 91.3%

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

   3. *-commutative 91.3%

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

   4. associate-*r* 98.8%

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

   5. +-commutative 98.8%

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

   6. +-commutative 98.8%

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

9. Applied egg-rr 98.8%

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

10. Final simplification 98.8%

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

Alternative 3: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;{x}^{-0.5} - \left(1 + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.1)
   (- (pow x -0.5) (+ 1.0 (* x -0.5)))
   (* (pow x -0.5) (- (/ 0.5 x) (/ 0.375 (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = pow(x, -0.5) - (1.0 + (x * -0.5));
	} else {
		tmp = pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.1d0) then
        tmp = (x ** (-0.5d0)) - (1.0d0 + (x * (-0.5d0)))
    else
        tmp = (x ** (-0.5d0)) * ((0.5d0 / x) - (0.375d0 / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = Math.pow(x, -0.5) - (1.0 + (x * -0.5));
	} else {
		tmp = Math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.1:
		tmp = math.pow(x, -0.5) - (1.0 + (x * -0.5))
	else:
		tmp = math.pow(x, -0.5) * ((0.5 / x) - (0.375 / (x * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.1)
		tmp = Float64((x ^ -0.5) - Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64((x ^ -0.5) * Float64(Float64(0.5 / x) - Float64(0.375 / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.1)
		tmp = (x ^ -0.5) - (1.0 + (x * -0.5));
	else
		tmp = (x ^ -0.5) * ((0.5 / x) - (0.375 / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.1], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x, -0.5], $MachinePrecision] * N[(N[(0.5 / x), $MachinePrecision] - N[(0.375 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around 0 98.8%

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

      1. add-log-exp 4.4%

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

      2. *-un-lft-identity 4.4%

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

      3. log-prod 4.4%

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

      4. metadata-eval 4.4%

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

      5. add-log-exp 98.2%

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

      6. pow1/2 98.2%

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

      7. pow-flip 98.5%

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

      8. metadata-eval 98.5%

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

   4. Applied egg-rr 99.2%

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

      1. +-lft-identity 98.5%

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

   6. Simplified 99.2%

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

   if 1.1000000000000001 < x

   1. Initial program 36.2%

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

      1. frac-sub 36.1%

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

      2. div-inv 36.1%

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

      3. *-un-lft-identity 36.1%

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

      4. +-commutative 36.1%

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

      5. *-rgt-identity 36.1%

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

      6. metadata-eval 36.1%

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

      7. frac-times 36.1%

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

      8. un-div-inv 36.1%

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

      9. pow1/2 36.1%

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

      10. pow-flip 36.1%

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

      11. metadata-eval 36.1%

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

      12. +-commutative 36.1%

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

   3. Applied egg-rr 36.1%

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

      1. associate-*r/ 36.1%

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

      2. *-rgt-identity 36.1%

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

      3. times-frac 36.1%

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

      4. div-sub 36.2%

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

      5. *-inverses 36.2%

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

      6. /-rgt-identity 36.2%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in x around inf 97.9%

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

      1. associate-*r/ 97.9%

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

      2. metadata-eval 97.9%

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

      3. associate-*r/ 97.9%

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

      4. metadata-eval 97.9%

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

      5. unpow2 97.9%

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

   8. Simplified 97.9%

      \[\leadsto \color{blue}{\left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)} \cdot {x}^{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;{x}^{-0.5} - \left(1 + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(\frac{0.5}{x} - \frac{0.375}{x \cdot x}\right)\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;{x}^{-0.5} - \left(1 + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.1)
   (- (pow x -0.5) (+ 1.0 (* x -0.5)))
   (/ (* (pow x -0.5) (+ 0.5 (/ -0.375 x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = pow(x, -0.5) - (1.0 + (x * -0.5));
	} else {
		tmp = (pow(x, -0.5) * (0.5 + (-0.375 / x))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.1d0) then
        tmp = (x ** (-0.5d0)) - (1.0d0 + (x * (-0.5d0)))
    else
        tmp = ((x ** (-0.5d0)) * (0.5d0 + ((-0.375d0) / x))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.1) {
		tmp = Math.pow(x, -0.5) - (1.0 + (x * -0.5));
	} else {
		tmp = (Math.pow(x, -0.5) * (0.5 + (-0.375 / x))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.1:
		tmp = math.pow(x, -0.5) - (1.0 + (x * -0.5))
	else:
		tmp = (math.pow(x, -0.5) * (0.5 + (-0.375 / x))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.1)
		tmp = Float64((x ^ -0.5) - Float64(1.0 + Float64(x * -0.5)));
	else
		tmp = Float64(Float64((x ^ -0.5) * Float64(0.5 + Float64(-0.375 / x))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.1)
		tmp = (x ^ -0.5) - (1.0 + (x * -0.5));
	else
		tmp = ((x ^ -0.5) * (0.5 + (-0.375 / x))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.1], N[(N[Power[x, -0.5], $MachinePrecision] - N[(1.0 + N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x, -0.5], $MachinePrecision] * N[(0.5 + N[(-0.375 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1000000000000001

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around 0 98.8%

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

      1. add-log-exp 4.4%

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

      2. *-un-lft-identity 4.4%

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

      3. log-prod 4.4%

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

      4. metadata-eval 4.4%

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

      5. add-log-exp 98.2%

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

      6. pow1/2 98.2%

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

      7. pow-flip 98.5%

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

      8. metadata-eval 98.5%

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

   4. Applied egg-rr 99.2%

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

      1. +-lft-identity 98.5%

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

   6. Simplified 99.2%

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

   if 1.1000000000000001 < x

   1. Initial program 36.2%

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

      1. frac-sub 36.1%

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

      2. div-inv 36.1%

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

      3. *-un-lft-identity 36.1%

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

      4. +-commutative 36.1%

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

      5. *-rgt-identity 36.1%

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

      6. metadata-eval 36.1%

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

      7. frac-times 36.1%

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

      8. un-div-inv 36.1%

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

      9. pow1/2 36.1%

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

      10. pow-flip 36.1%

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

      11. metadata-eval 36.1%

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

      12. +-commutative 36.1%

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

   3. Applied egg-rr 36.1%

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

      1. associate-*r/ 36.1%

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

      2. *-rgt-identity 36.1%

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

      3. times-frac 36.1%

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

      4. div-sub 36.2%

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

      5. *-inverses 36.2%

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

      6. /-rgt-identity 36.2%

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

   5. Simplified 36.2%

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

      1. flip-- 36.2%

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

      2. metadata-eval 36.2%

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

      3. sqrt-undiv 36.3%

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

      4. sqrt-undiv 36.2%

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

      5. add-sqr-sqrt 36.4%

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

      6. sqrt-undiv 36.4%

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

   7. Applied egg-rr 36.4%

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

   8. Taylor expanded in x around inf 97.9%

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

      1. associate-*r/ 97.9%

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

      2. metadata-eval 97.9%

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

      3. associate-*r/ 97.9%

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

      4. metadata-eval 97.9%

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

      5. unpow2 97.9%

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

      6. associate-/r* 97.9%

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

   10. Simplified 97.9%

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

       1. *-commutative 97.9%

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

       2. sub-div 97.9%

          \[\leadsto {x}^{-0.5} \cdot \color{blue}{\frac{0.5 - \frac{0.375}{x}}{x}} \]

       3. associate-*r/ 97.9%

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

       4. sub-neg 97.9%

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

       5. distribute-neg-frac 97.9%

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

       6. metadata-eval 97.9%

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

   12. Applied egg-rr 97.9%

       \[\leadsto \color{blue}{\frac{{x}^{-0.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1:\\ \;\;\;\;{x}^{-0.5} - \left(1 + x \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-0.5} \cdot \left(0.5 + \frac{-0.375}{x}\right)}{x}\\ \end{array} \]

Alternative 5: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around 0 98.8%

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

      1. add-log-exp 4.4%

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

      2. *-un-lft-identity 4.4%

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

      3. log-prod 4.4%

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

      4. metadata-eval 4.4%

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

      5. add-log-exp 98.2%

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

      6. pow1/2 98.2%

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

      7. pow-flip 98.5%

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

      8. metadata-eval 98.5%

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

   4. Applied egg-rr 99.2%

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

      1. +-lft-identity 98.5%

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

   6. Simplified 99.2%

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

   if 1 < x

   1. Initial program 36.2%

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

      1. frac-sub 36.1%

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

      2. div-inv 36.1%

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

      3. *-un-lft-identity 36.1%

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

      4. +-commutative 36.1%

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

      5. *-rgt-identity 36.1%

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

      6. metadata-eval 36.1%

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

      7. frac-times 36.1%

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

      8. un-div-inv 36.1%

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

      9. pow1/2 36.1%

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

      10. pow-flip 36.1%

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

      11. metadata-eval 36.1%

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

      12. +-commutative 36.1%

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

   3. Applied egg-rr 36.1%

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

      1. associate-*r/ 36.1%

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

      2. *-rgt-identity 36.1%

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

      3. times-frac 36.1%

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

      4. div-sub 36.2%

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

      5. *-inverses 36.2%

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

      6. /-rgt-identity 36.2%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in x around inf 97.2%

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

      1. expm1-log1p-u 97.2%

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

      2. expm1-udef 33.8%

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

      3. div-inv 33.8%

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

      4. associate-*l* 33.8%

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

      5. inv-pow 33.8%

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

      6. metadata-eval 33.8%

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

      7. pow-prod-up 33.8%

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

      8. pow3 33.8%

         \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{{\left({x}^{-0.5}\right)}^{3}}\right)} - 1 \]

      9. pow-pow 33.8%

         \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{{x}^{\left(-0.5 \cdot 3\right)}}\right)} - 1 \]

      10. metadata-eval 33.8%

          \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot {x}^{\color{blue}{-1.5}}\right)} - 1 \]

   8. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 97.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)\right)} \]

      2. expm1-log1p 97.6%

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

   10. Simplified 97.6%

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

4. Final simplification 98.4%

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

Alternative 6: 97.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 99.6%

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

      1. frac-sub 99.6%

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

      2. div-inv 99.6%

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

      3. *-un-lft-identity 99.6%

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

      4. +-commutative 99.6%

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

      5. *-rgt-identity 99.6%

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

      6. metadata-eval 99.6%

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

      7. frac-times 99.6%

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

      8. un-div-inv 99.6%

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

      9. pow1/2 99.6%

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

      10. pow-flip 100.0%

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

      11. metadata-eval 100.0%

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

      12. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. times-frac 100.0%

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

      4. div-sub 100.0%

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

      5. *-inverses 100.0%

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

      6. /-rgt-identity 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 94.8%

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

   if 0.5 < x

   1. Initial program 36.2%

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

      1. frac-sub 36.1%

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

      2. div-inv 36.1%

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

      3. *-un-lft-identity 36.1%

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

      4. +-commutative 36.1%

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

      5. *-rgt-identity 36.1%

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

      6. metadata-eval 36.1%

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

      7. frac-times 36.1%

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

      8. un-div-inv 36.1%

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

      9. pow1/2 36.1%

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

      10. pow-flip 36.1%

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

      11. metadata-eval 36.1%

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

      12. +-commutative 36.1%

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

   3. Applied egg-rr 36.1%

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

      1. associate-*r/ 36.1%

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

      2. *-rgt-identity 36.1%

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

      3. times-frac 36.1%

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

      4. div-sub 36.2%

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

      5. *-inverses 36.2%

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

      6. /-rgt-identity 36.2%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in x around inf 97.2%

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

      1. expm1-log1p-u 97.2%

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

      2. expm1-udef 33.8%

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

      3. div-inv 33.8%

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

      4. associate-*l* 33.8%

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

      5. inv-pow 33.8%

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

      6. metadata-eval 33.8%

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

      7. pow-prod-up 33.8%

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

      8. pow3 33.8%

         \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{{\left({x}^{-0.5}\right)}^{3}}\right)} - 1 \]

      9. pow-pow 33.8%

         \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{{x}^{\left(-0.5 \cdot 3\right)}}\right)} - 1 \]

      10. metadata-eval 33.8%

          \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot {x}^{\color{blue}{-1.5}}\right)} - 1 \]

   8. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 97.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)\right)} \]

      2. expm1-log1p 97.6%

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

   10. Simplified 97.6%

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

4. Final simplification 96.2%

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

Alternative 7: 98.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.680000000000000049

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

   2. Taylor expanded in x around 0 98.2%

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

      1. add-log-exp 4.4%

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

      2. *-un-lft-identity 4.4%

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

      3. log-prod 4.4%

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

      4. metadata-eval 4.4%

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

      5. add-log-exp 98.2%

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

      6. pow1/2 98.2%

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

      7. pow-flip 98.5%

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

      8. metadata-eval 98.5%

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

   4. Applied egg-rr 98.5%

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

      1. +-lft-identity 98.5%

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

   6. Simplified 98.5%

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

   if 0.680000000000000049 < x

   1. Initial program 36.2%

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

      1. frac-sub 36.1%

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

      2. div-inv 36.1%

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

      3. *-un-lft-identity 36.1%

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

      4. +-commutative 36.1%

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

      5. *-rgt-identity 36.1%

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

      6. metadata-eval 36.1%

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

      7. frac-times 36.1%

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

      8. un-div-inv 36.1%

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

      9. pow1/2 36.1%

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

      10. pow-flip 36.1%

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

      11. metadata-eval 36.1%

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

      12. +-commutative 36.1%

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

   3. Applied egg-rr 36.1%

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

      1. associate-*r/ 36.1%

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

      2. *-rgt-identity 36.1%

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

      3. times-frac 36.1%

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

      4. div-sub 36.2%

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

      5. *-inverses 36.2%

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

      6. /-rgt-identity 36.2%

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

   5. Simplified 36.2%

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

   6. Taylor expanded in x around inf 97.2%

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

      1. expm1-log1p-u 97.2%

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

      2. expm1-udef 33.8%

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

      3. div-inv 33.8%

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

      4. associate-*l* 33.8%

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

      5. inv-pow 33.8%

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

      6. metadata-eval 33.8%

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

      7. pow-prod-up 33.8%

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

      8. pow3 33.8%

         \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{{\left({x}^{-0.5}\right)}^{3}}\right)} - 1 \]

      9. pow-pow 33.8%

         \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{{x}^{\left(-0.5 \cdot 3\right)}}\right)} - 1 \]

      10. metadata-eval 33.8%

          \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot {x}^{\color{blue}{-1.5}}\right)} - 1 \]

   8. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 97.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)\right)} \]

      2. expm1-log1p 97.6%

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

   10. Simplified 97.6%

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

4. Final simplification 98.0%

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

Alternative 8: 51.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot {x}^{-1.5} \end{array} \]

FPCore

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

C

double code(double x) {
	return 0.5 * pow(x, -1.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 * (x ** (-1.5d0))
end function

Java

public static double code(double x) {
	return 0.5 * Math.pow(x, -1.5);
}

Python

def code(x):
	return 0.5 * math.pow(x, -1.5)

Julia

function code(x)
	return Float64(0.5 * (x ^ -1.5))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 * (x ^ -1.5);
end

Wolfram

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

Derivation

1. Initial program 67.7%

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

   1. frac-sub 67.6%

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

   2. div-inv 67.6%

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

   3. *-un-lft-identity 67.6%

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

   4. +-commutative 67.6%

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

   5. *-rgt-identity 67.6%

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

   6. metadata-eval 67.6%

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

   7. frac-times 67.6%

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

   8. un-div-inv 67.6%

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

   9. pow1/2 67.6%

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

   10. pow-flip 67.8%

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

   11. metadata-eval 67.8%

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

   12. +-commutative 67.8%

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

3. Applied egg-rr 67.8%

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

   1. associate-*r/ 67.8%

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

   2. *-rgt-identity 67.8%

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

   3. times-frac 67.8%

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

   4. div-sub 67.8%

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

   5. *-inverses 67.8%

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

   6. /-rgt-identity 67.8%

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

5. Simplified 67.8%

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

6. Taylor expanded in x around inf 51.8%

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

   1. expm1-log1p-u 51.8%

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

   2. expm1-udef 19.9%

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

   3. div-inv 19.9%

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

   4. associate-*l* 19.9%

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

   5. inv-pow 19.9%

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

   6. metadata-eval 19.9%

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

   7. pow-prod-up 19.9%

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

   8. pow3 19.9%

      \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{{\left({x}^{-0.5}\right)}^{3}}\right)} - 1 \]

   9. pow-pow 19.9%

      \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot \color{blue}{{x}^{\left(-0.5 \cdot 3\right)}}\right)} - 1 \]

   10. metadata-eval 19.9%

       \[\leadsto e^{\mathsf{log1p}\left(0.5 \cdot {x}^{\color{blue}{-1.5}}\right)} - 1 \]

8. Applied egg-rr 19.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)} - 1} \]
9. Step-by-step derivation

   1. expm1-def 52.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot {x}^{-1.5}\right)\right)} \]

   2. expm1-log1p 52.0%

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

10. Simplified 52.0%

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

11. Final simplification 52.0%

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

Alternative 9: 3.9% accurate, 69.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x 0.5))

C

double code(double x) {
	return 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 x * 0.5;
}

Python

def code(x):
	return x * 0.5

Julia

function code(x)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.5;
end

Wolfram

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

Derivation

1. Initial program 67.7%

   \[\frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x + 1}} \]

2. Taylor expanded in x around 0 50.8%

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

3. Taylor expanded in x around inf 4.0%

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

   1. *-commutative 4.0%

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

5. Simplified 4.0%

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

6. Final simplification 4.0%

   \[\leadsto x \cdot 0.5 \]

Developer target: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023238 
(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)))))