Main:bigenough3 from C

Percentage Accurate: 53.7% → 99.8%

Time: 6.3s

Alternatives: 9

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: 53.7% 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: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt{x}, x, {\left(1 + x\right)}^{1.5}\right)} \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\left(1 + x\right) \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 0.0)
   (* (sqrt (pow x -1.0)) 0.5)
   (*
    (/ (- (+ 1.0 x) x) (fma (sqrt x) x (pow (+ 1.0 x) 1.5)))
    (- (+ (+ 1.0 x) x) (sqrt (* (+ 1.0 x) x))))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 0.0) {
		tmp = sqrt(pow(x, -1.0)) * 0.5;
	} else {
		tmp = (((1.0 + x) - x) / fma(sqrt(x), x, pow((1.0 + x), 1.5))) * (((1.0 + x) + x) - sqrt(((1.0 + x) * x)));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision] - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(1.0 + x), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] * x + N[Power[N[(1.0 + x), $MachinePrecision], 1.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(1.0 + x), $MachinePrecision] + x), $MachinePrecision] - N[Sqrt[N[(N[(1.0 + x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0

   1. Initial program 3.9%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 98.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. pow-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. rem-log-exp N/A

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

      11. pow-to-exp N/A

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

      12. log-pow N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. metadata-eval 98.9

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

   4. Applied rewrites 98.9%

      \[\leadsto \sqrt{x + 1} - \color{blue}{e^{\left(2 \cdot \log x\right) \cdot 0.25}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\sqrt{x + 1} - e^{\left(2 \cdot \log x\right) \cdot \frac{1}{4}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{x + 1}} - e^{\left(2 \cdot \log x\right) \cdot \frac{1}{4}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{x + 1}} - e^{\left(2 \cdot \log x\right) \cdot \frac{1}{4}} \]

      4. lift-exp.f64 N/A

         \[\leadsto \sqrt{x + 1} - \color{blue}{e^{\left(2 \cdot \log x\right) \cdot \frac{1}{4}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{x + 1} - e^{\color{blue}{\left(2 \cdot \log x\right) \cdot \frac{1}{4}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{x + 1} - e^{\color{blue}{\left(2 \cdot \log x\right)} \cdot \frac{1}{4}} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{x + 1} - e^{\color{blue}{\left(\log x \cdot 2\right)} \cdot \frac{1}{4}} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{x + 1} - e^{\color{blue}{\log x \cdot \left(2 \cdot \frac{1}{4}\right)}} \]

      9. lift-log.f64 N/A

         \[\leadsto \sqrt{x + 1} - e^{\color{blue}{\log x} \cdot \left(2 \cdot \frac{1}{4}\right)} \]

      10. metadata-eval N/A

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

      11. pow-to-exp N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. flip-- N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip3-+ N/A

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

      4. associate-/r/ N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x\right) - x}{\mathsf{fma}\left(\sqrt{x}, x, {\left(1 + x\right)}^{1.5}\right)} \cdot \left(\left(\left(1 + x\right) + x\right) - \sqrt{\left(1 + x\right) \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1}\\ \mathbf{if}\;t\_0 - \sqrt{x} \leq 0:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\sqrt{x} + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (+ x 1.0))))
   (if (<= (- t_0 (sqrt x)) 0.0)
     (* (sqrt (pow x -1.0)) 0.5)
     (/ (- (+ x 1.0) x) (+ (sqrt x) t_0)))))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0));
	double tmp;
	if ((t_0 - sqrt(x)) <= 0.0) {
		tmp = sqrt(pow(x, -1.0)) * 0.5;
	} else {
		tmp = ((x + 1.0) - x) / (sqrt(x) + 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))
    if ((t_0 - sqrt(x)) <= 0.0d0) then
        tmp = sqrt((x ** (-1.0d0))) * 0.5d0
    else
        tmp = ((x + 1.0d0) - x) / (sqrt(x) + t_0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.sqrt((x + 1.0));
	double tmp;
	if ((t_0 - Math.sqrt(x)) <= 0.0) {
		tmp = Math.sqrt(Math.pow(x, -1.0)) * 0.5;
	} else {
		tmp = ((x + 1.0) - x) / (Math.sqrt(x) + t_0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.sqrt((x + 1.0))
	tmp = 0
	if (t_0 - math.sqrt(x)) <= 0.0:
		tmp = math.sqrt(math.pow(x, -1.0)) * 0.5
	else:
		tmp = ((x + 1.0) - x) / (math.sqrt(x) + t_0)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[(x + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 - N[Sqrt[x], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - x), $MachinePrecision] / N[(N[Sqrt[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x)) < 0.0

   1. Initial program 3.9%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 0.0 < (-.f64 (sqrt.f64 (+.f64 x #s(literal 1 binary64))) (sqrt.f64 x))

   1. Initial program 98.9%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. sqr-pow N/A

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

      4. pow-prod-down N/A

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

      5. pow2 N/A

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

      6. pow-to-exp N/A

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

      7. pow-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. rem-log-exp N/A

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

      11. pow-to-exp N/A

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

      12. log-pow N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. metadata-eval 98.9

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

   4. Applied rewrites 98.9%

      \[\leadsto \sqrt{x + 1} - \color{blue}{e^{\left(2 \cdot \log x\right) \cdot 0.25}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\sqrt{x + 1} - e^{\left(2 \cdot \log x\right) \cdot \frac{1}{4}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{x + 1}} - e^{\left(2 \cdot \log x\right) \cdot \frac{1}{4}} \]

      3. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{x + 1}} - e^{\left(2 \cdot \log x\right) \cdot \frac{1}{4}} \]

      4. lift-exp.f64 N/A

         \[\leadsto \sqrt{x + 1} - \color{blue}{e^{\left(2 \cdot \log x\right) \cdot \frac{1}{4}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{x + 1} - e^{\color{blue}{\left(2 \cdot \log x\right) \cdot \frac{1}{4}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{x + 1} - e^{\color{blue}{\left(2 \cdot \log x\right)} \cdot \frac{1}{4}} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{x + 1} - e^{\color{blue}{\left(\log x \cdot 2\right)} \cdot \frac{1}{4}} \]

      8. associate-*l* N/A

         \[\leadsto \sqrt{x + 1} - e^{\color{blue}{\log x \cdot \left(2 \cdot \frac{1}{4}\right)}} \]

      9. lift-log.f64 N/A

         \[\leadsto \sqrt{x + 1} - e^{\color{blue}{\log x} \cdot \left(2 \cdot \frac{1}{4}\right)} \]

      10. metadata-eval N/A

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

      11. pow-to-exp N/A

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

      12. pow1/2 N/A

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

      13. lift-sqrt.f64 N/A

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

      14. flip-- N/A

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

      15. lower-/.f64 N/A

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 0:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - x}{\sqrt{x} + \sqrt{x + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x + 1} - \sqrt{x}\\ \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (sqrt (+ x 1.0)) (sqrt x))))
   (if (<= t_0 2e-6) (* (sqrt (pow x -1.0)) 0.5) t_0)))

C

double code(double x) {
	double t_0 = sqrt((x + 1.0)) - sqrt(x);
	double tmp;
	if (t_0 <= 2e-6) {
		tmp = sqrt(pow(x, -1.0)) * 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 <= 2d-6) then
        tmp = sqrt((x ** (-1.0d0))) * 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 <= 2e-6) {
		tmp = Math.sqrt(Math.pow(x, -1.0)) * 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 <= 2e-6:
		tmp = math.sqrt(math.pow(x, -1.0)) * 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 <= 2e-6)
		tmp = Float64(sqrt((x ^ -1.0)) * 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 <= 2e-6)
		tmp = sqrt((x ^ -1.0)) * 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, 2e-6], N[(N[Sqrt[N[Power[x, -1.0], $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)) < 1.99999999999999991e-6

   1. Initial program 4.3%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 1.99999999999999991e-6 < (-.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. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x + 1} - \sqrt{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 2e-6)
   (* (sqrt (pow x -1.0)) 0.5)
   (fma (fma -0.125 x 0.5) x (- 1.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 2e-6) {
		tmp = sqrt(pow(x, -1.0)) * 0.5;
	} else {
		tmp = fma(fma(-0.125, x, 0.5), x, (1.0 - sqrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 2e-6)
		tmp = Float64(sqrt((x ^ -1.0)) * 0.5);
	else
		tmp = fma(fma(-0.125, x, 0.5), x, 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], 2e-6], N[(N[Sqrt[N[Power[x, -1.0], $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x + 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)) < 1.99999999999999991e-6

   1. Initial program 4.3%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if 1.99999999999999991e-6 < (-.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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{{x}^{-1}} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.125, x, 0.5\right), x, 1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 2e-6)
   (/ 0.5 (sqrt x))
   (fma (fma -0.125 x 0.5) x (- 1.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 2e-6) {
		tmp = 0.5 / sqrt(x);
	} else {
		tmp = fma(fma(-0.125, x, 0.5), x, (1.0 - sqrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 2e-6)
		tmp = Float64(0.5 / sqrt(x));
	else
		tmp = fma(fma(-0.125, x, 0.5), x, 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], 2e-6], N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[(-0.125 * x + 0.5), $MachinePrecision] * x + 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)) < 1.99999999999999991e-6

   1. Initial program 4.3%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.4%

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

   if 1.99999999999999991e-6 < (-.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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 6: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x + 1} - \sqrt{x} \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{0.5}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, x, 1 - \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (sqrt (+ x 1.0)) (sqrt x)) 2e-6)
   (/ 0.5 (sqrt x))
   (fma 0.5 x (- 1.0 (sqrt x)))))

C

double code(double x) {
	double tmp;
	if ((sqrt((x + 1.0)) - sqrt(x)) <= 2e-6) {
		tmp = 0.5 / sqrt(x);
	} else {
		tmp = fma(0.5, x, (1.0 - sqrt(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(sqrt(Float64(x + 1.0)) - sqrt(x)) <= 2e-6)
		tmp = Float64(0.5 / sqrt(x));
	else
		tmp = fma(0.5, x, 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], 2e-6], N[(0.5 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * x + 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)) < 1.99999999999999991e-6

   1. Initial program 4.3%

      \[\sqrt{x + 1} - \sqrt{x} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.4%

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

   if 1.99999999999999991e-6 < (-.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

      \[\leadsto \color{blue}{\left(1 + \frac{1}{2} \cdot x\right) - \sqrt{x}} \]
   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. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-sqrt.f64 98.9

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

   5. Applied rewrites 98.9%

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

Alternative 7: 52.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-sqrt.f64 51.5

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

5. Applied rewrites 51.5%

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

Alternative 8: 50.2% 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 52.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 49.8%

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

Alternative 9: 1.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot -0.125 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* x x) -0.125))

C

double code(double x) {
	return (x * x) * -0.125;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * (-0.125d0)
end function

Java

public static double code(double x) {
	return (x * x) * -0.125;
}

Python

def code(x):
	return (x * x) * -0.125

Julia

function code(x)
	return Float64(Float64(x * x) * -0.125)
end

MATLAB

function tmp = code(x)
	tmp = (x * x) * -0.125;
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * -0.125), $MachinePrecision]

Derivation

1. Initial program 52.1%

   \[\sqrt{x + 1} - \sqrt{x} \]
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. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-sqrt.f64 50.2

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

5. Applied rewrites 50.2%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 1.9%

   \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{-0.125} \]
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 2024320 
(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)))