Main:bigenough3 from C

Percentage Accurate: 52.8% → 99.7%

Time: 8.6s

Alternatives: 11

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(-0.0390625, \sqrt{\frac{1}{{x}^{5}}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \sqrt{x} \cdot 0.5\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 0.01)
     (/
      (fma
       -0.125
       (sqrt (/ 1.0 x))
       (fma
        -0.0390625
        (sqrt (/ 1.0 (pow x 5.0)))
        (fma 0.0625 (sqrt (/ 1.0 (* x (* x x)))) (* (sqrt x) 0.5))))
      x)
     t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = fma(-0.125, sqrt((1.0 / x)), fma(-0.0390625, sqrt((1.0 / pow(x, 5.0))), fma(0.0625, sqrt((1.0 / (x * (x * x)))), (sqrt(x) * 0.5)))) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = Float64(fma(-0.125, sqrt(Float64(1.0 / x)), fma(-0.0390625, sqrt(Float64(1.0 / (x ^ 5.0))), fma(0.0625, sqrt(Float64(1.0 / Float64(x * Float64(x * x)))), Float64(sqrt(x) * 0.5)))) / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.01], N[(N[(-0.125 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(-0.0390625 * N[Sqrt[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(0.0625 * N[Sqrt[N[(1.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0100000000000000002

   1. Initial program 6.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{-5}{128} \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{-5}{128} \cdot \sqrt{\frac{1}{{x}^{5}}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x}} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(-0.0390625, \sqrt{\frac{1}{{x}^{5}}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \sqrt{x} \cdot 0.5\right)\right)\right)}{x}} \]

   if 0.0100000000000000002 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.8%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{{x}^{5}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 0.01)
     (fma
      -0.125
      (sqrt (/ 1.0 (* x (* x x))))
      (fma 0.0625 (sqrt (/ 1.0 (pow x 5.0))) (* (sqrt (/ 1.0 x)) 0.5)))
     t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = fma(-0.125, sqrt((1.0 / (x * (x * x)))), fma(0.0625, sqrt((1.0 / pow(x, 5.0))), (sqrt((1.0 / x)) * 0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = fma(-0.125, sqrt(Float64(1.0 / Float64(x * Float64(x * x)))), fma(0.0625, sqrt(Float64(1.0 / (x ^ 5.0))), Float64(sqrt(Float64(1.0 / x)) * 0.5)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.01], N[(-0.125 * N[Sqrt[N[(1.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(0.0625 * N[Sqrt[N[(1.0 / N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0100000000000000002

   1. Initial program 6.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}}{x} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{\sqrt{\frac{1}{x}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\color{blue}{\frac{1}{x}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \color{blue}{\mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{{x}^{3}}}, \frac{1}{2} \cdot \sqrt{x}\right)}\right)}{x} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \color{blue}{\sqrt{\frac{1}{{x}^{3}}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\color{blue}{\frac{1}{{x}^{3}}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

      8. cube-mult N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{x \cdot \color{blue}{\left(x \cdot x\right)}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\sqrt{x} \cdot \frac{1}{2}}\right)\right)}{x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\sqrt{x} \cdot \frac{1}{2}}\right)\right)}{x} \]

      13. lower-sqrt.f64 99.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\sqrt{x}} \cdot 0.5\right)\right)}{x} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \sqrt{x} \cdot 0.5\right)\right)}{x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{5}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{{x}^{3}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{5}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \color{blue}{\sqrt{\frac{1}{{x}^{3}}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{5}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\color{blue}{\frac{1}{{x}^{3}}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{5}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) \]

      4. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{5}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x \cdot \color{blue}{{x}^{2}}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{5}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{\color{blue}{x \cdot {x}^{2}}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{5}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x \cdot \color{blue}{\left(x \cdot x\right)}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{5}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x \cdot \color{blue}{\left(x \cdot x\right)}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{5}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{{x}^{5}}}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)}\right) \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(\frac{1}{16}, \color{blue}{\sqrt{\frac{1}{{x}^{5}}}}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\color{blue}{\frac{1}{{x}^{5}}}}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right) \]

      12. lower-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\color{blue}{{x}^{5}}}}, \frac{1}{2} \cdot \sqrt{\frac{1}{x}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{{x}^{5}}}, \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}}\right)\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{{x}^{5}}}, \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}}\right)\right) \]

      15. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{{x}^{5}}}, \color{blue}{\sqrt{\frac{1}{x}}} \cdot \frac{1}{2}\right)\right) \]

      16. lower-/.f64 99.6

          \[\leadsto \mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{{x}^{5}}}, \sqrt{\color{blue}{\frac{1}{x}}} \cdot 0.5\right)\right) \]

   8. Simplified 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{{x}^{5}}}, \sqrt{\frac{1}{x}} \cdot 0.5\right)\right)} \]

   if 0.0100000000000000002 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.8%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, -0.125 \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 5e-5)
     (fma (sqrt (/ 1.0 x)) 0.5 (* -0.125 (sqrt (/ 1.0 (* x (* x x))))))
     t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = fma(sqrt((1.0 / x)), 0.5, (-0.125 * sqrt((1.0 / (x * (x * x))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 5e-5)
		tmp = fma(sqrt(Float64(1.0 / x)), 0.5, Float64(-0.125 * sqrt(Float64(1.0 / Float64(x * Float64(x * x))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-5], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5 + N[(-0.125 * N[Sqrt[N[(1.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 5.00000000000000024e-5

   1. Initial program 5.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \left(\frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}}{x} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{\sqrt{\frac{1}{x}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\color{blue}{\frac{1}{x}}}, \frac{1}{16} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{x}\right)}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \color{blue}{\mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{{x}^{3}}}, \frac{1}{2} \cdot \sqrt{x}\right)}\right)}{x} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \color{blue}{\sqrt{\frac{1}{{x}^{3}}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\color{blue}{\frac{1}{{x}^{3}}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

      8. cube-mult N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{x \cdot \color{blue}{\left(x \cdot x\right)}}}, \frac{1}{2} \cdot \sqrt{x}\right)\right)}{x} \]

      11. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\sqrt{x} \cdot \frac{1}{2}}\right)\right)}{x} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(\frac{1}{16}, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\sqrt{x} \cdot \frac{1}{2}}\right)\right)}{x} \]

      13. lower-sqrt.f64 99.6

          \[\leadsto \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \color{blue}{\sqrt{x}} \cdot 0.5\right)\right)}{x} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \mathsf{fma}\left(0.0625, \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}, \sqrt{x} \cdot 0.5\right)\right)}{x}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} + \frac{1}{2} \cdot \sqrt{\frac{1}{x}}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \sqrt{\frac{1}{x}} + \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \frac{1}{2}} + \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{2}, \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}\right)} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{x}}}, \frac{1}{2}, \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{x}}}, \frac{1}{2}, \frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{2}, \color{blue}{\frac{-1}{8} \cdot \sqrt{\frac{1}{{x}^{3}}}}\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{2}, \frac{-1}{8} \cdot \color{blue}{\sqrt{\frac{1}{{x}^{3}}}}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{2}, \frac{-1}{8} \cdot \sqrt{\color{blue}{\frac{1}{{x}^{3}}}}\right) \]

      9. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{2}, \frac{-1}{8} \cdot \sqrt{\frac{1}{\color{blue}{x \cdot \left(x \cdot x\right)}}}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{2}, \frac{-1}{8} \cdot \sqrt{\frac{1}{x \cdot \color{blue}{{x}^{2}}}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{2}, \frac{-1}{8} \cdot \sqrt{\frac{1}{\color{blue}{x \cdot {x}^{2}}}}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, \frac{1}{2}, \frac{-1}{8} \cdot \sqrt{\frac{1}{x \cdot \color{blue}{\left(x \cdot x\right)}}}\right) \]

      13. lower-*.f64 99.8

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, -0.125 \cdot \sqrt{\frac{1}{x \cdot \color{blue}{\left(x \cdot x\right)}}}\right) \]

   8. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{x}}, 0.5, -0.125 \cdot \sqrt{\frac{1}{x \cdot \left(x \cdot x\right)}}\right)} \]

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.3%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \sqrt{x} \cdot 0.5\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 5e-5)
     (/ (fma -0.125 (sqrt (/ 1.0 x)) (* (sqrt x) 0.5)) x)
     t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = fma(-0.125, sqrt((1.0 / x)), (sqrt(x) * 0.5)) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 5e-5)
		tmp = Float64(fma(-0.125, sqrt(Float64(1.0 / x)), Float64(sqrt(x) * 0.5)) / x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-5], N[(N[(-0.125 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 5.00000000000000024e-5

   1. Initial program 5.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \sqrt{\frac{1}{x}} + \frac{1}{2} \cdot \sqrt{x}}{x}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \frac{1}{2} \cdot \sqrt{x}\right)}}{x} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \color{blue}{\sqrt{\frac{1}{x}}}, \frac{1}{2} \cdot \sqrt{x}\right)}{x} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\color{blue}{\frac{1}{x}}}, \frac{1}{2} \cdot \sqrt{x}\right)}{x} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{x} \cdot \frac{1}{2}}\right)}{x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{8}, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{x} \cdot \frac{1}{2}}\right)}{x} \]

      7. lower-sqrt.f64 99.6

         \[\leadsto \frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \color{blue}{\sqrt{x}} \cdot 0.5\right)}{x} \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.125, \sqrt{\frac{1}{x}}, \sqrt{x} \cdot 0.5\right)}{x}} \]

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.3%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 5e-5) (* (sqrt (/ 1.0 x)) 0.5) t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = sqrt((1.0 / x)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x + 1.0d0)) - sqrt(x)
    if (t_0 <= 5d-5) then
        tmp = sqrt((1.0d0 / x)) * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0)) - Math.sqrt(x);
	double tmp;
	if (t_0 <= 5e-5) {
		tmp = Math.sqrt((1.0 / x)) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0)) - math.sqrt(x)
	tmp = 0
	if t_0 <= 5e-5:
		tmp = math.sqrt((1.0 / x)) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(sqrt(Float64(x + 1.0)) - sqrt(x))
	tmp = 0.0
	if (t_0 <= 5e-5)
		tmp = Float64(sqrt(Float64(1.0 / x)) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = sqrt((x + 1.0)) - sqrt(x);
	tmp = 0.0;
	if (t_0 <= 5e-5)
		tmp = sqrt((1.0 / x)) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-5], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 5.00000000000000024e-5

   1. Initial program 5.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 99.1

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

   5. Simplified 99.1%

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

   if 5.00000000000000024e-5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 99.3%

      \[\sqrt{x + 1} - \sqrt{x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

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

Alternative 6: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.5:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.5)
   (* (sqrt (/ 1.0 x)) 0.5)
   (fma x (fma x -0.125 0.5) (- 1.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.5) {
		tmp = sqrt((1.0 / x)) * 0.5;
	} else {
		tmp = fma(x, fma(x, -0.125, 0.5), (1.0 - sqrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 0.5)
		tmp = Float64(sqrt(Float64(1.0 / x)) * 0.5);
	else
		tmp = fma(x, fma(x, -0.125, 0.5), Float64(1.0 - sqrt(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.5], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(x * N[(x * -0.125 + 0.5), $MachinePrecision] + N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.5

   1. Initial program 9.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 95.7

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

   5. Simplified 95.7%

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

   if 0.5 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right)\right) - \sqrt{x}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + 1\right)} - \sqrt{x} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1}{2} + \frac{-1}{8} \cdot x\right) + \left(1 - \sqrt{x}\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{2} + \frac{-1}{8} \cdot x, 1 - \sqrt{x}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\frac{-1}{8} \cdot x + \frac{1}{2}}, 1 - \sqrt{x}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{8}} + \frac{1}{2}, 1 - \sqrt{x}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right)}, 1 - \sqrt{x}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{-1}{8}, \frac{1}{2}\right), \color{blue}{1 - \sqrt{x}}\right) \]

      8. lower-sqrt.f64 99.0

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1 - \color{blue}{\sqrt{x}}\right) \]

   5. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1 - \sqrt{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.5:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.125, 0.5\right), 1 - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0.2:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, -\sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.2)
   (fma x 0.5 (- (sqrt x)))
   (- 1.0 (sqrt x))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.2) {
		tmp = fma(x, 0.5, -sqrt(x));
	} else {
		tmp = 1.0 - sqrt(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 0.2)
		tmp = fma(x, 0.5, Float64(-sqrt(x)));
	else
		tmp = Float64(1.0 - sqrt(x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.2], N[(x * 0.5 + (-N[Sqrt[x], $MachinePrecision])), $MachinePrecision], N[(1.0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.20000000000000001

   1. Initial program 8.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-sqrt.f64 4.8

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

   5. Simplified 4.8%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-sqrt.f64 4.8

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

   8. Simplified 4.8%

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

   if 0.20000000000000001 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-sqrt.f64 97.0

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

   5. Simplified 97.0%

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

Alternative 8: 51.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 0.5 (- 1.0 (sqrt x))))

C

double code(double x) {
	return fma(x, 0.5, (1.0 - sqrt(x)));
}

Julia

function code(x)
	return fma(x, 0.5, Float64(1.0 - sqrt(x)))
end

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-sqrt.f64 50.7

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

5. Simplified 50.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1 - \sqrt{x}\right)} \]
6. Add Preprocessing

Alternative 9: 51.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.5, 1\right) - \sqrt{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (fma x 0.5 1.0) (sqrt x)))

C

double code(double x) {
	return fma(x, 0.5, 1.0) - sqrt(x);
}

Julia

function code(x)
	return Float64(fma(x, 0.5, 1.0) - sqrt(x))
end

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 50.7

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

5. Simplified 50.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)} - \sqrt{x} \]
6. Add Preprocessing

Alternative 10: 49.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-sqrt.f64 48.5

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

5. Simplified 48.5%

   \[\leadsto \color{blue}{1 - \sqrt{x}} \]
6. Add Preprocessing

Alternative 11: 3.5% accurate, 27.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

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

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 53.6%

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

3. Taylor expanded in x around inf

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

   1. lower-sqrt.f64 3.5

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

5. Simplified 3.5%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{0} \]
7. Step-by-step derivation

8. Simplified 3.5%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "Main:bigenough3 from C"
  :precision binary64

  :alt
  (! :herbie-platform default (/ 1 (+ (sqrt (+ x 1)) (sqrt x))))

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