Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B

Percentage Accurate: 99.4% → 99.4%

Time: 11.2s

Alternatives: 13

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))

C

double code(double x, double y) {
	return (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * ((y + (1.0d0 / (x * 9.0d0))) - 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0)

Julia

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
end

Wolfram

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))

C

double code(double x, double y) {
	return (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * ((y + (1.0d0 / (x * 9.0d0))) - 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0)

Julia

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
end

Wolfram

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ t_0 \cdot \left({t_0}^{-2} + \left(y + -1\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (* x 9.0)))) (* t_0 (+ (pow t_0 -2.0) (+ y -1.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt((x * 9.0));
	return t_0 * (pow(t_0, -2.0) + (y + -1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = sqrt((x * 9.0d0))
    code = t_0 * ((t_0 ** (-2.0d0)) + (y + (-1.0d0)))
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((x * 9.0));
	return t_0 * (Math.pow(t_0, -2.0) + (y + -1.0));
}

Python

def code(x, y):
	t_0 = math.sqrt((x * 9.0))
	return t_0 * (math.pow(t_0, -2.0) + (y + -1.0))

Julia

function code(x, y)
	t_0 = sqrt(Float64(x * 9.0))
	return Float64(t_0 * Float64((t_0 ^ -2.0) + Float64(y + -1.0)))
end

MATLAB

function tmp = code(x, y)
	t_0 = sqrt((x * 9.0));
	tmp = t_0 * ((t_0 ^ -2.0) + (y + -1.0));
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]}, N[(t$95$0 * N[(N[Power[t$95$0, -2.0], $MachinePrecision] + N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   2. associate-*l* 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

   3. sub-neg 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

   4. +-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right)\right) \]

   5. +-lft-identity 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(\frac{1}{x \cdot 9} + \color{blue}{\left(0 + y\right)}\right) + \left(-1\right)\right)\right) \]

   6. associate-+l+ 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)}\right) \]

   7. *-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

   8. associate-/r* 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

   9. metadata-eval 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

   10. +-lft-identity 99.4%

       \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(\color{blue}{y} + \left(-1\right)\right)\right)\right) \]

   11. metadata-eval 99.4%

       \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + \color{blue}{-1}\right)\right)\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]
4. Step-by-step derivation

   1. metadata-eval 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{\frac{1}{9}}}{x} + \left(y + -1\right)\right)\right) \]

   2. associate-/r* 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{1}{9 \cdot x}} + \left(y + -1\right)\right)\right) \]

   3. *-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{x \cdot 9}} + \left(y + -1\right)\right)\right) \]

   4. associate-+l+ 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(\frac{1}{x \cdot 9} + y\right) + -1\right)}\right) \]

   5. +-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} + -1\right)\right) \]

   6. metadata-eval 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + \color{blue}{\left(-1\right)}\right)\right) \]

   7. sub-neg 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)}\right) \]

   8. associate-*l* 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)} \]

   9. *-commutative 99.4%

      \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   10. associate--l+ 99.4%

       \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)} \]

   11. distribute-lft-in 99.4%

       \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right)} \]

   12. *-commutative 99.4%

       \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

   13. metadata-eval 99.4%

       \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

   14. sqrt-prod 99.5%

       \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

   15. *-commutative 99.5%

       \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

   16. metadata-eval 99.5%

       \[\leadsto \sqrt{x \cdot 9} \cdot y + \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

   17. sqrt-prod 99.6%

       \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

5. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot y + \sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \color{blue}{y \cdot \sqrt{x \cdot 9}} + \sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   2. distribute-lft-in 99.5%

      \[\leadsto y \cdot \sqrt{x \cdot 9} + \color{blue}{\left(\sqrt{x \cdot 9} \cdot \frac{0.1111111111111111}{x} + \sqrt{x \cdot 9} \cdot -1\right)} \]

   3. *-commutative 99.5%

      \[\leadsto y \cdot \sqrt{x \cdot 9} + \left(\color{blue}{\frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9}} + \sqrt{x \cdot 9} \cdot -1\right) \]

   4. +-commutative 99.5%

      \[\leadsto y \cdot \sqrt{x \cdot 9} + \color{blue}{\left(\sqrt{x \cdot 9} \cdot -1 + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9}\right)} \]

   5. associate-+l+ 99.5%

      \[\leadsto \color{blue}{\left(y \cdot \sqrt{x \cdot 9} + \sqrt{x \cdot 9} \cdot -1\right) + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9}} \]

   6. *-commutative 99.5%

      \[\leadsto \left(\color{blue}{\sqrt{x \cdot 9} \cdot y} + \sqrt{x \cdot 9} \cdot -1\right) + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9} \]

   7. distribute-lft-in 99.5%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(y + -1\right)} + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9} \]

   8. *-commutative 99.5%

      \[\leadsto \color{blue}{\left(y + -1\right) \cdot \sqrt{x \cdot 9}} + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9} \]

   9. distribute-rgt-out 99.5%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(\left(y + -1\right) + \frac{0.1111111111111111}{x}\right)} \]

   10. +-commutative 99.5%

       \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)} \]

7. Simplified 99.5%

   \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)} \]
8. Step-by-step derivation

   1. metadata-eval 99.5%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{\color{blue}{\frac{1}{9}}}{x} + \left(y + -1\right)\right) \]

   2. associate-/r* 99.6%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\frac{1}{9 \cdot x}} + \left(y + -1\right)\right) \]

   3. *-commutative 99.6%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{1}{\color{blue}{x \cdot 9}} + \left(y + -1\right)\right) \]

   4. inv-pow 99.6%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{{\left(x \cdot 9\right)}^{-1}} + \left(y + -1\right)\right) \]

   5. *-commutative 99.6%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left({\color{blue}{\left(9 \cdot x\right)}}^{-1} + \left(y + -1\right)\right) \]

   6. metadata-eval 99.6%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left({\left(\color{blue}{\left(3 \cdot 3\right)} \cdot x\right)}^{-1} + \left(y + -1\right)\right) \]

   7. add-sqr-sqrt 99.4%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left({\left(\left(3 \cdot 3\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right)}^{-1} + \left(y + -1\right)\right) \]

   8. swap-sqr 99.4%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left({\color{blue}{\left(\left(3 \cdot \sqrt{x}\right) \cdot \left(3 \cdot \sqrt{x}\right)\right)}}^{-1} + \left(y + -1\right)\right) \]

   9. unpow-prod-down 99.3%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{{\left(3 \cdot \sqrt{x}\right)}^{-1} \cdot {\left(3 \cdot \sqrt{x}\right)}^{-1}} + \left(y + -1\right)\right) \]

   10. *-commutative 99.3%

       \[\leadsto \sqrt{x \cdot 9} \cdot \left({\color{blue}{\left(\sqrt{x} \cdot 3\right)}}^{-1} \cdot {\left(3 \cdot \sqrt{x}\right)}^{-1} + \left(y + -1\right)\right) \]

   11. metadata-eval 99.3%

       \[\leadsto \sqrt{x \cdot 9} \cdot \left({\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right)}^{-1} \cdot {\left(3 \cdot \sqrt{x}\right)}^{-1} + \left(y + -1\right)\right) \]

   12. sqrt-prod 99.4%

       \[\leadsto \sqrt{x \cdot 9} \cdot \left({\color{blue}{\left(\sqrt{x \cdot 9}\right)}}^{-1} \cdot {\left(3 \cdot \sqrt{x}\right)}^{-1} + \left(y + -1\right)\right) \]

   13. *-commutative 99.4%

       \[\leadsto \sqrt{x \cdot 9} \cdot \left({\left(\sqrt{x \cdot 9}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{x} \cdot 3\right)}}^{-1} + \left(y + -1\right)\right) \]

   14. metadata-eval 99.4%

       \[\leadsto \sqrt{x \cdot 9} \cdot \left({\left(\sqrt{x \cdot 9}\right)}^{-1} \cdot {\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right)}^{-1} + \left(y + -1\right)\right) \]

   15. sqrt-prod 99.4%

       \[\leadsto \sqrt{x \cdot 9} \cdot \left({\left(\sqrt{x \cdot 9}\right)}^{-1} \cdot {\color{blue}{\left(\sqrt{x \cdot 9}\right)}}^{-1} + \left(y + -1\right)\right) \]

9. Applied egg-rr 99.4%

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

    1. pow-sqr 99.6%

       \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{{\left(\sqrt{x \cdot 9}\right)}^{\left(2 \cdot -1\right)}} + \left(y + -1\right)\right) \]

    2. metadata-eval 99.6%

       \[\leadsto \sqrt{x \cdot 9} \cdot \left({\left(\sqrt{x \cdot 9}\right)}^{\color{blue}{-2}} + \left(y + -1\right)\right) \]

11. Simplified 99.6%

    \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{{\left(\sqrt{x \cdot 9}\right)}^{-2}} + \left(y + -1\right)\right) \]

12. Final simplification 99.6%

    \[\leadsto \sqrt{x \cdot 9} \cdot \left({\left(\sqrt{x \cdot 9}\right)}^{-2} + \left(y + -1\right)\right) \]

Alternative 2: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ t_1 := 3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{if}\;x \leq 1.6 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 0.00078:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+70} \lor \neg \left(x \leq 5 \cdot 10^{+138}\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x))) (t_1 (* 3.0 (* y (sqrt x)))))
   (if (<= x 1.6e-109)
     t_0
     (if (<= x 4e-102)
       t_1
       (if (<= x 0.00078)
         t_0
         (if (or (<= x 2.5e+70) (not (<= x 5e+138)))
           (* (sqrt x) -3.0)
           t_1))))))

C

double code(double x, double y) {
	double t_0 = sqrt((0.1111111111111111 / x));
	double t_1 = 3.0 * (y * sqrt(x));
	double tmp;
	if (x <= 1.6e-109) {
		tmp = t_0;
	} else if (x <= 4e-102) {
		tmp = t_1;
	} else if (x <= 0.00078) {
		tmp = t_0;
	} else if ((x <= 2.5e+70) || !(x <= 5e+138)) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    t_1 = 3.0d0 * (y * sqrt(x))
    if (x <= 1.6d-109) then
        tmp = t_0
    else if (x <= 4d-102) then
        tmp = t_1
    else if (x <= 0.00078d0) then
        tmp = t_0
    else if ((x <= 2.5d+70) .or. (.not. (x <= 5d+138))) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((0.1111111111111111 / x));
	double t_1 = 3.0 * (y * Math.sqrt(x));
	double tmp;
	if (x <= 1.6e-109) {
		tmp = t_0;
	} else if (x <= 4e-102) {
		tmp = t_1;
	} else if (x <= 0.00078) {
		tmp = t_0;
	} else if ((x <= 2.5e+70) || !(x <= 5e+138)) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((0.1111111111111111 / x))
	t_1 = 3.0 * (y * math.sqrt(x))
	tmp = 0
	if x <= 1.6e-109:
		tmp = t_0
	elif x <= 4e-102:
		tmp = t_1
	elif x <= 0.00078:
		tmp = t_0
	elif (x <= 2.5e+70) or not (x <= 5e+138):
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(0.1111111111111111 / x))
	t_1 = Float64(3.0 * Float64(y * sqrt(x)))
	tmp = 0.0
	if (x <= 1.6e-109)
		tmp = t_0;
	elseif (x <= 4e-102)
		tmp = t_1;
	elseif (x <= 0.00078)
		tmp = t_0;
	elseif ((x <= 2.5e+70) || !(x <= 5e+138))
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((0.1111111111111111 / x));
	t_1 = 3.0 * (y * sqrt(x));
	tmp = 0.0;
	if (x <= 1.6e-109)
		tmp = t_0;
	elseif (x <= 4e-102)
		tmp = t_1;
	elseif (x <= 0.00078)
		tmp = t_0;
	elseif ((x <= 2.5e+70) || ~((x <= 5e+138)))
		tmp = sqrt(x) * -3.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.6e-109], t$95$0, If[LessEqual[x, 4e-102], t$95$1, If[LessEqual[x, 0.00078], t$95$0, If[Or[LessEqual[x, 2.5e+70], N[Not[LessEqual[x, 5e+138]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.6000000000000001e-109 or 3.99999999999999973e-102 < x < 7.79999999999999986e-4

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-udef 7.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      3. *-commutative 7.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot 3}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      4. metadata-eval 7.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. sqrt-prod 7.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   3. Applied egg-rr 7.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   4. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   6. Step-by-step derivation

      1. sub-neg 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)} \]

      2. +-commutative 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right) \]

      3. metadata-eval 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\left(\frac{1}{x \cdot 9} + y\right) + \color{blue}{-1}\right) \]

      4. associate-+l+ 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(y + -1\right)\right)} \]

      5. *-commutative 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(y + -1\right)\right) \]

      6. associate-/r* 99.3%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(y + -1\right)\right) \]

      7. metadata-eval 99.3%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(y + -1\right)\right) \]

      8. add-sqr-sqrt 87.7%

         \[\leadsto \color{blue}{\sqrt{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)} \cdot \sqrt{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}} \]

      9. sqrt-unprod 82.9%

         \[\leadsto \color{blue}{\sqrt{\left(\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right) \cdot \left(\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)}} \]

      10. swap-sqr 35.0%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x \cdot 9} \cdot \sqrt{x \cdot 9}\right) \cdot \left(\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)}} \]

      11. add-sqr-sqrt 35.1%

          \[\leadsto \sqrt{\color{blue}{\left(x \cdot 9\right)} \cdot \left(\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]

      12. pow2 35.1%

          \[\leadsto \sqrt{\left(x \cdot 9\right) \cdot \color{blue}{{\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}^{2}}} \]

   7. Applied egg-rr 35.1%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot 9\right) \cdot {\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}^{2}}} \]

   8. Taylor expanded in x around 0 78.3%

      \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \]

   if 1.6000000000000001e-109 < x < 3.99999999999999973e-102 or 2.5000000000000001e70 < x < 5.00000000000000016e138

   1. Initial program 99.2%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. associate-*l* 99.4%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      3. sub-neg 99.4%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

      4. +-commutative 99.4%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right)\right) \]

      5. +-lft-identity 99.4%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(\frac{1}{x \cdot 9} + \color{blue}{\left(0 + y\right)}\right) + \left(-1\right)\right)\right) \]

      6. associate-+l+ 99.4%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)}\right) \]

      7. *-commutative 99.4%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      8. associate-/r* 99.4%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      9. metadata-eval 99.4%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      10. +-lft-identity 99.4%

          \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(\color{blue}{y} + \left(-1\right)\right)\right)\right) \]

      11. metadata-eval 99.4%

          \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + \color{blue}{-1}\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]

   4. Taylor expanded in y around inf 68.8%

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

   if 7.79999999999999986e-4 < x < 2.5000000000000001e70 or 5.00000000000000016e138 < x

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. associate-*l* 99.6%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      3. sub-neg 99.6%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

      4. +-commutative 99.6%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(-1\right) + \left(y + \frac{1}{x \cdot 9}\right)\right)}\right) \]

      5. distribute-lft-in 99.6%

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(-1\right) + 3 \cdot \left(y + \frac{1}{x \cdot 9}\right)\right)} \]

      6. metadata-eval 99.6%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{-1} + 3 \cdot \left(y + \frac{1}{x \cdot 9}\right)\right) \]

      7. metadata-eval 99.6%

         \[\leadsto \sqrt{x} \cdot \left(\color{blue}{-3} + 3 \cdot \left(y + \frac{1}{x \cdot 9}\right)\right) \]

      8. distribute-lft-in 99.6%

         \[\leadsto \sqrt{x} \cdot \left(-3 + \color{blue}{\left(3 \cdot y + 3 \cdot \frac{1}{x \cdot 9}\right)}\right) \]

      9. *-commutative 99.6%

         \[\leadsto \sqrt{x} \cdot \left(-3 + \left(3 \cdot y + \color{blue}{\frac{1}{x \cdot 9} \cdot 3}\right)\right) \]

      10. fma-def 99.6%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \color{blue}{\mathsf{fma}\left(3, y, \frac{1}{x \cdot 9} \cdot 3\right)}\right) \]

      11. *-commutative 99.6%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{1}{\color{blue}{9 \cdot x}} \cdot 3\right)\right) \]

      12. associate-/r* 99.6%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \color{blue}{\frac{\frac{1}{9}}{x}} \cdot 3\right)\right) \]

      13. associate-*l/ 99.6%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \color{blue}{\frac{\frac{1}{9} \cdot 3}{x}}\right)\right) \]

      14. metadata-eval 99.6%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{\color{blue}{0.1111111111111111} \cdot 3}{x}\right)\right) \]

      15. metadata-eval 99.6%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{\color{blue}{0.3333333333333333}}{x}\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\right)} \]

   4. Taylor expanded in x around inf 98.6%

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

   5. Taylor expanded in y around 0 60.8%

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

      1. *-commutative 60.8%

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

   7. Simplified 60.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-102}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 0.00078:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+70} \lor \neg \left(x \leq 5 \cdot 10^{+138}\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \]

Alternative 3: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{if}\;x \leq 2.65 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-102}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x))))
   (if (<= x 2.65e-109)
     t_0
     (if (<= x 4e-102)
       (* 3.0 (* y (sqrt x)))
       (if (<= x 1.75e-19) t_0 (* (sqrt x) (- (* y 3.0) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt((0.1111111111111111 / x));
	double tmp;
	if (x <= 2.65e-109) {
		tmp = t_0;
	} else if (x <= 4e-102) {
		tmp = 3.0 * (y * sqrt(x));
	} else if (x <= 1.75e-19) {
		tmp = t_0;
	} else {
		tmp = sqrt(x) * ((y * 3.0) - 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    if (x <= 2.65d-109) then
        tmp = t_0
    else if (x <= 4d-102) then
        tmp = 3.0d0 * (y * sqrt(x))
    else if (x <= 1.75d-19) then
        tmp = t_0
    else
        tmp = sqrt(x) * ((y * 3.0d0) - 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((0.1111111111111111 / x));
	double tmp;
	if (x <= 2.65e-109) {
		tmp = t_0;
	} else if (x <= 4e-102) {
		tmp = 3.0 * (y * Math.sqrt(x));
	} else if (x <= 1.75e-19) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(x) * ((y * 3.0) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((0.1111111111111111 / x))
	tmp = 0
	if x <= 2.65e-109:
		tmp = t_0
	elif x <= 4e-102:
		tmp = 3.0 * (y * math.sqrt(x))
	elif x <= 1.75e-19:
		tmp = t_0
	else:
		tmp = math.sqrt(x) * ((y * 3.0) - 3.0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(0.1111111111111111 / x))
	tmp = 0.0
	if (x <= 2.65e-109)
		tmp = t_0;
	elseif (x <= 4e-102)
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	elseif (x <= 1.75e-19)
		tmp = t_0;
	else
		tmp = Float64(sqrt(x) * Float64(Float64(y * 3.0) - 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((0.1111111111111111 / x));
	tmp = 0.0;
	if (x <= 2.65e-109)
		tmp = t_0;
	elseif (x <= 4e-102)
		tmp = 3.0 * (y * sqrt(x));
	elseif (x <= 1.75e-19)
		tmp = t_0;
	else
		tmp = sqrt(x) * ((y * 3.0) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.65e-109], t$95$0, If[LessEqual[x, 4e-102], N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.75e-19], t$95$0, N[(N[Sqrt[x], $MachinePrecision] * N[(N[(y * 3.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.6499999999999999e-109 or 3.99999999999999973e-102 < x < 1.75000000000000008e-19

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-udef 4.5%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      3. *-commutative 4.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot 3}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      4. metadata-eval 4.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. sqrt-prod 4.5%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   3. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   4. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   6. Step-by-step derivation

      1. sub-neg 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)} \]

      2. +-commutative 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right) \]

      3. metadata-eval 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\left(\frac{1}{x \cdot 9} + y\right) + \color{blue}{-1}\right) \]

      4. associate-+l+ 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(y + -1\right)\right)} \]

      5. *-commutative 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(y + -1\right)\right) \]

      6. associate-/r* 99.3%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(y + -1\right)\right) \]

      7. metadata-eval 99.3%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(y + -1\right)\right) \]

      8. add-sqr-sqrt 89.7%

         \[\leadsto \color{blue}{\sqrt{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)} \cdot \sqrt{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}} \]

      9. sqrt-unprod 84.7%

         \[\leadsto \color{blue}{\sqrt{\left(\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right) \cdot \left(\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)}} \]

      10. swap-sqr 34.8%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x \cdot 9} \cdot \sqrt{x \cdot 9}\right) \cdot \left(\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)}} \]

      11. add-sqr-sqrt 34.9%

          \[\leadsto \sqrt{\color{blue}{\left(x \cdot 9\right)} \cdot \left(\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]

      12. pow2 34.9%

          \[\leadsto \sqrt{\left(x \cdot 9\right) \cdot \color{blue}{{\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}^{2}}} \]

   7. Applied egg-rr 34.9%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot 9\right) \cdot {\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}^{2}}} \]

   8. Taylor expanded in x around 0 80.6%

      \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \]

   if 2.6499999999999999e-109 < x < 3.99999999999999973e-102

   1. Initial program 98.8%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. associate-*l* 99.7%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      3. sub-neg 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

      4. +-commutative 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right)\right) \]

      5. +-lft-identity 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(\frac{1}{x \cdot 9} + \color{blue}{\left(0 + y\right)}\right) + \left(-1\right)\right)\right) \]

      6. associate-+l+ 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)}\right) \]

      7. *-commutative 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      8. associate-/r* 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      9. metadata-eval 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      10. +-lft-identity 99.7%

          \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(\color{blue}{y} + \left(-1\right)\right)\right)\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + \color{blue}{-1}\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]

   4. Taylor expanded in y around inf 99.7%

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

   if 1.75000000000000008e-19 < x

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. associate-*l* 99.5%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      3. sub-neg 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

      4. +-commutative 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(-1\right) + \left(y + \frac{1}{x \cdot 9}\right)\right)}\right) \]

      5. distribute-lft-in 99.5%

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(-1\right) + 3 \cdot \left(y + \frac{1}{x \cdot 9}\right)\right)} \]

      6. metadata-eval 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{-1} + 3 \cdot \left(y + \frac{1}{x \cdot 9}\right)\right) \]

      7. metadata-eval 99.5%

         \[\leadsto \sqrt{x} \cdot \left(\color{blue}{-3} + 3 \cdot \left(y + \frac{1}{x \cdot 9}\right)\right) \]

      8. distribute-lft-in 99.5%

         \[\leadsto \sqrt{x} \cdot \left(-3 + \color{blue}{\left(3 \cdot y + 3 \cdot \frac{1}{x \cdot 9}\right)}\right) \]

      9. *-commutative 99.5%

         \[\leadsto \sqrt{x} \cdot \left(-3 + \left(3 \cdot y + \color{blue}{\frac{1}{x \cdot 9} \cdot 3}\right)\right) \]

      10. fma-def 99.5%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \color{blue}{\mathsf{fma}\left(3, y, \frac{1}{x \cdot 9} \cdot 3\right)}\right) \]

      11. *-commutative 99.5%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{1}{\color{blue}{9 \cdot x}} \cdot 3\right)\right) \]

      12. associate-/r* 99.5%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \color{blue}{\frac{\frac{1}{9}}{x}} \cdot 3\right)\right) \]

      13. associate-*l/ 99.5%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \color{blue}{\frac{\frac{1}{9} \cdot 3}{x}}\right)\right) \]

      14. metadata-eval 99.5%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{\color{blue}{0.1111111111111111} \cdot 3}{x}\right)\right) \]

      15. metadata-eval 99.5%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{\color{blue}{0.3333333333333333}}{x}\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\right)} \]

   4. Taylor expanded in x around inf 97.4%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot y - 3\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-102}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-19}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \end{array} \]

Alternative 4: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{if}\;x \leq 2.65 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-102}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x))))
   (if (<= x 2.65e-109)
     t_0
     (if (<= x 4e-102)
       (* 3.0 (* y (sqrt x)))
       (if (<= x 1.2e-21) t_0 (* (sqrt (* x 9.0)) (+ y -1.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt((0.1111111111111111 / x));
	double tmp;
	if (x <= 2.65e-109) {
		tmp = t_0;
	} else if (x <= 4e-102) {
		tmp = 3.0 * (y * sqrt(x));
	} else if (x <= 1.2e-21) {
		tmp = t_0;
	} else {
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    if (x <= 2.65d-109) then
        tmp = t_0
    else if (x <= 4d-102) then
        tmp = 3.0d0 * (y * sqrt(x))
    else if (x <= 1.2d-21) then
        tmp = t_0
    else
        tmp = sqrt((x * 9.0d0)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((0.1111111111111111 / x));
	double tmp;
	if (x <= 2.65e-109) {
		tmp = t_0;
	} else if (x <= 4e-102) {
		tmp = 3.0 * (y * Math.sqrt(x));
	} else if (x <= 1.2e-21) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((0.1111111111111111 / x))
	tmp = 0
	if x <= 2.65e-109:
		tmp = t_0
	elif x <= 4e-102:
		tmp = 3.0 * (y * math.sqrt(x))
	elif x <= 1.2e-21:
		tmp = t_0
	else:
		tmp = math.sqrt((x * 9.0)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(0.1111111111111111 / x))
	tmp = 0.0
	if (x <= 2.65e-109)
		tmp = t_0;
	elseif (x <= 4e-102)
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	elseif (x <= 1.2e-21)
		tmp = t_0;
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((0.1111111111111111 / x));
	tmp = 0.0;
	if (x <= 2.65e-109)
		tmp = t_0;
	elseif (x <= 4e-102)
		tmp = 3.0 * (y * sqrt(x));
	elseif (x <= 1.2e-21)
		tmp = t_0;
	else
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.65e-109], t$95$0, If[LessEqual[x, 4e-102], N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.2e-21], t$95$0, N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.6499999999999999e-109 or 3.99999999999999973e-102 < x < 1.2e-21

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-udef 3.7%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      3. *-commutative 3.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot 3}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      4. metadata-eval 3.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. sqrt-prod 3.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   3. Applied egg-rr 3.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   4. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   6. Step-by-step derivation

      1. sub-neg 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)} \]

      2. +-commutative 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right) \]

      3. metadata-eval 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\left(\frac{1}{x \cdot 9} + y\right) + \color{blue}{-1}\right) \]

      4. associate-+l+ 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(y + -1\right)\right)} \]

      5. *-commutative 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(y + -1\right)\right) \]

      6. associate-/r* 99.3%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(y + -1\right)\right) \]

      7. metadata-eval 99.3%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(y + -1\right)\right) \]

      8. add-sqr-sqrt 90.4%

         \[\leadsto \color{blue}{\sqrt{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)} \cdot \sqrt{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}} \]

      9. sqrt-unprod 85.3%

         \[\leadsto \color{blue}{\sqrt{\left(\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right) \cdot \left(\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)}} \]

      10. swap-sqr 34.6%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x \cdot 9} \cdot \sqrt{x \cdot 9}\right) \cdot \left(\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)}} \]

      11. add-sqr-sqrt 34.6%

          \[\leadsto \sqrt{\color{blue}{\left(x \cdot 9\right)} \cdot \left(\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]

      12. pow2 34.6%

          \[\leadsto \sqrt{\left(x \cdot 9\right) \cdot \color{blue}{{\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}^{2}}} \]

   7. Applied egg-rr 34.6%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot 9\right) \cdot {\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}^{2}}} \]

   8. Taylor expanded in x around 0 81.1%

      \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \]

   if 2.6499999999999999e-109 < x < 3.99999999999999973e-102

   1. Initial program 98.8%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative 98.8%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. associate-*l* 99.7%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      3. sub-neg 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

      4. +-commutative 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right)\right) \]

      5. +-lft-identity 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(\frac{1}{x \cdot 9} + \color{blue}{\left(0 + y\right)}\right) + \left(-1\right)\right)\right) \]

      6. associate-+l+ 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)}\right) \]

      7. *-commutative 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      8. associate-/r* 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      9. metadata-eval 99.7%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      10. +-lft-identity 99.7%

          \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(\color{blue}{y} + \left(-1\right)\right)\right)\right) \]

      11. metadata-eval 99.7%

          \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + \color{blue}{-1}\right)\right)\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]

   4. Taylor expanded in y around inf 99.7%

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

   if 1.2e-21 < x

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 93.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-udef 92.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      3. *-commutative 92.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot 3}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      4. metadata-eval 92.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. sqrt-prod 92.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   3. Applied egg-rr 92.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   4. Step-by-step derivation

      1. expm1-def 93.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-log1p 99.6%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   6. Taylor expanded in y around inf 96.7%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{y} - 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-102}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot 9} \cdot \left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (sqrt (* x 9.0)) (+ -1.0 (+ y (/ 1.0 (* x 9.0))))))

C

double code(double x, double y) {
	return sqrt((x * 9.0)) * (-1.0 + (y + (1.0 / (x * 9.0))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((x * 9.0d0)) * ((-1.0d0) + (y + (1.0d0 / (x * 9.0d0))))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt((x * 9.0)) * (-1.0 + (y + (1.0 / (x * 9.0))));
}

Python

def code(x, y):
	return math.sqrt((x * 9.0)) * (-1.0 + (y + (1.0 / (x * 9.0))))

Julia

function code(x, y)
	return Float64(sqrt(Float64(x * 9.0)) * Float64(-1.0 + Float64(y + Float64(1.0 / Float64(x * 9.0)))))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((x * 9.0)) * (-1.0 + (y + (1.0 / (x * 9.0))));
end

Wolfram

code[x_, y_] := N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(-1.0 + N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Step-by-step derivation

   1. expm1-log1p-u 96.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   2. expm1-udef 49.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   3. *-commutative 49.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot 3}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   4. metadata-eval 49.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. sqrt-prod 49.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

3. Applied egg-rr 49.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
4. Step-by-step derivation

   1. expm1-def 96.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   2. expm1-log1p 99.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

6. Final simplification 99.6%

   \[\leadsto \sqrt{x \cdot 9} \cdot \left(-1 + \left(y + \frac{1}{x \cdot 9}\right)\right) \]

Alternative 6: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+14} \lor \neg \left(y \leq 4.4 \cdot 10^{+101}\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.7e+14) (not (<= y 4.4e+101)))
   (* (sqrt (* x 9.0)) y)
   (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.7e+14) || !(y <= 4.4e+101)) {
		tmp = sqrt((x * 9.0)) * y;
	} else {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.7d+14)) .or. (.not. (y <= 4.4d+101))) then
        tmp = sqrt((x * 9.0d0)) * y
    else
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.7e+14) || !(y <= 4.4e+101)) {
		tmp = Math.sqrt((x * 9.0)) * y;
	} else {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.7e+14) or not (y <= 4.4e+101):
		tmp = math.sqrt((x * 9.0)) * y
	else:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.7e+14) || !(y <= 4.4e+101))
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	else
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.7e+14) || ~((y <= 4.4e+101)))
		tmp = sqrt((x * 9.0)) * y;
	else
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.7e+14], N[Not[LessEqual[y, 4.4e+101]], $MachinePrecision]], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7e14 or 4.4000000000000001e101 < y

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. associate-*l* 99.5%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      3. sub-neg 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

      4. +-commutative 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right)\right) \]

      5. +-lft-identity 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(\frac{1}{x \cdot 9} + \color{blue}{\left(0 + y\right)}\right) + \left(-1\right)\right)\right) \]

      6. associate-+l+ 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)}\right) \]

      7. *-commutative 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      8. associate-/r* 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      9. metadata-eval 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      10. +-lft-identity 99.5%

          \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(\color{blue}{y} + \left(-1\right)\right)\right)\right) \]

      11. metadata-eval 99.5%

          \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + \color{blue}{-1}\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]
   4. Step-by-step derivation

      1. metadata-eval 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{\frac{1}{9}}}{x} + \left(y + -1\right)\right)\right) \]

      2. associate-/r* 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{1}{9 \cdot x}} + \left(y + -1\right)\right)\right) \]

      3. *-commutative 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{x \cdot 9}} + \left(y + -1\right)\right)\right) \]

      4. associate-+l+ 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(\frac{1}{x \cdot 9} + y\right) + -1\right)}\right) \]

      5. +-commutative 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} + -1\right)\right) \]

      6. metadata-eval 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + \color{blue}{\left(-1\right)}\right)\right) \]

      7. sub-neg 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)}\right) \]

      8. associate-*l* 99.5%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)} \]

      9. *-commutative 99.5%

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      10. associate--l+ 99.5%

          \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)} \]

      11. distribute-lft-in 99.5%

          \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right)} \]

      12. *-commutative 99.5%

          \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

      13. metadata-eval 99.5%

          \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

      14. sqrt-prod 99.6%

          \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

      15. *-commutative 99.6%

          \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \sqrt{x \cdot 9} \cdot y + \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

      17. sqrt-prod 99.6%

          \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot y + \sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)} \]
   6. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{y \cdot \sqrt{x \cdot 9}} + \sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      2. distribute-lft-in 99.6%

         \[\leadsto y \cdot \sqrt{x \cdot 9} + \color{blue}{\left(\sqrt{x \cdot 9} \cdot \frac{0.1111111111111111}{x} + \sqrt{x \cdot 9} \cdot -1\right)} \]

      3. *-commutative 99.6%

         \[\leadsto y \cdot \sqrt{x \cdot 9} + \left(\color{blue}{\frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9}} + \sqrt{x \cdot 9} \cdot -1\right) \]

      4. +-commutative 99.6%

         \[\leadsto y \cdot \sqrt{x \cdot 9} + \color{blue}{\left(\sqrt{x \cdot 9} \cdot -1 + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9}\right)} \]

      5. associate-+l+ 99.6%

         \[\leadsto \color{blue}{\left(y \cdot \sqrt{x \cdot 9} + \sqrt{x \cdot 9} \cdot -1\right) + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9}} \]

      6. *-commutative 99.6%

         \[\leadsto \left(\color{blue}{\sqrt{x \cdot 9} \cdot y} + \sqrt{x \cdot 9} \cdot -1\right) + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9} \]

      7. distribute-lft-in 99.6%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(y + -1\right)} + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9} \]

      8. *-commutative 99.6%

         \[\leadsto \color{blue}{\left(y + -1\right) \cdot \sqrt{x \cdot 9}} + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9} \]

      9. distribute-rgt-out 99.6%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(\left(y + -1\right) + \frac{0.1111111111111111}{x}\right)} \]

      10. +-commutative 99.6%

          \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)} \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)} \]

   8. Taylor expanded in y around inf 80.5%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{y} \]

   if -2.7e14 < y < 4.4000000000000001e101

   1. Initial program 99.4%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. associate-*l* 99.4%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      3. sub-neg 99.4%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

      4. +-commutative 99.4%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right)\right) \]

      5. +-lft-identity 99.4%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(\frac{1}{x \cdot 9} + \color{blue}{\left(0 + y\right)}\right) + \left(-1\right)\right)\right) \]

      6. associate-+l+ 99.4%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)}\right) \]

      7. *-commutative 99.4%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      8. associate-/r* 99.3%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      9. metadata-eval 99.3%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      10. +-lft-identity 99.3%

          \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(\color{blue}{y} + \left(-1\right)\right)\right)\right) \]

      11. metadata-eval 99.3%

          \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + \color{blue}{-1}\right)\right)\right) \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]

   4. Taylor expanded in y around 0 91.9%

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative 91.9%

         \[\leadsto 3 \cdot \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \sqrt{x}\right)} \]

      2. associate-*r/ 91.9%

         \[\leadsto 3 \cdot \left(\left(\color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} - 1\right) \cdot \sqrt{x}\right) \]

      3. metadata-eval 91.9%

         \[\leadsto 3 \cdot \left(\left(\frac{\color{blue}{0.1111111111111111}}{x} - 1\right) \cdot \sqrt{x}\right) \]

      4. sub-neg 91.9%

         \[\leadsto 3 \cdot \left(\color{blue}{\left(\frac{0.1111111111111111}{x} + \left(-1\right)\right)} \cdot \sqrt{x}\right) \]

      5. metadata-eval 91.9%

         \[\leadsto 3 \cdot \left(\left(\frac{0.1111111111111111}{x} + \color{blue}{-1}\right) \cdot \sqrt{x}\right) \]

      6. associate-*r* 91.9%

         \[\leadsto \color{blue}{\left(3 \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \sqrt{x}} \]

      7. distribute-lft-in 91.9%

         \[\leadsto \color{blue}{\left(3 \cdot \frac{0.1111111111111111}{x} + 3 \cdot -1\right)} \cdot \sqrt{x} \]

      8. associate-*r/ 92.0%

         \[\leadsto \left(\color{blue}{\frac{3 \cdot 0.1111111111111111}{x}} + 3 \cdot -1\right) \cdot \sqrt{x} \]

      9. metadata-eval 92.0%

         \[\leadsto \left(\frac{\color{blue}{0.3333333333333333}}{x} + 3 \cdot -1\right) \cdot \sqrt{x} \]

      10. metadata-eval 92.0%

          \[\leadsto \left(\frac{0.3333333333333333}{x} + \color{blue}{-3}\right) \cdot \sqrt{x} \]

      11. *-commutative 92.0%

          \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)} \]

   6. Simplified 92.0%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+14} \lor \neg \left(y \leq 4.4 \cdot 10^{+101}\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 7: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00078:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00078)
   (* (sqrt x) (+ (/ 0.3333333333333333 x) (* y 3.0)))
   (* (sqrt (* x 9.0)) (+ y -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00078) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + (y * 3.0));
	} else {
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.00078d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (y * 3.0d0))
    else
        tmp = sqrt((x * 9.0d0)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00078) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + (y * 3.0));
	} else {
		tmp = Math.sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00078:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + (y * 3.0))
	else:
		tmp = math.sqrt((x * 9.0)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00078)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + Float64(y * 3.0)));
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00078)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + (y * 3.0));
	else
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00078], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.79999999999999986e-4

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative 99.3%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. associate-*l* 99.3%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      3. sub-neg 99.3%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

      4. +-commutative 99.3%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right)\right) \]

      5. +-lft-identity 99.3%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(\frac{1}{x \cdot 9} + \color{blue}{\left(0 + y\right)}\right) + \left(-1\right)\right)\right) \]

      6. associate-+l+ 99.3%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)}\right) \]

      7. *-commutative 99.3%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      8. associate-/r* 99.2%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      9. metadata-eval 99.2%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

      10. +-lft-identity 99.2%

          \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(\color{blue}{y} + \left(-1\right)\right)\right)\right) \]

      11. metadata-eval 99.2%

          \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + \color{blue}{-1}\right)\right)\right) \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in 99.3%

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \frac{0.1111111111111111}{x} + 3 \cdot \left(y + -1\right)\right)} \]

      2. associate-*r/ 99.4%

         \[\leadsto \sqrt{x} \cdot \left(\color{blue}{\frac{3 \cdot 0.1111111111111111}{x}} + 3 \cdot \left(y + -1\right)\right) \]

      3. metadata-eval 99.4%

         \[\leadsto \sqrt{x} \cdot \left(\frac{\color{blue}{0.3333333333333333}}{x} + 3 \cdot \left(y + -1\right)\right) \]

      4. +-commutative 99.4%

         \[\leadsto \sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + 3 \cdot \color{blue}{\left(-1 + y\right)}\right) \]

      5. distribute-lft-in 99.4%

         \[\leadsto \sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + \color{blue}{\left(3 \cdot -1 + 3 \cdot y\right)}\right) \]

      6. metadata-eval 99.4%

         \[\leadsto \sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + \left(\color{blue}{-3} + 3 \cdot y\right)\right) \]

   5. Applied egg-rr 99.4%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(\frac{0.3333333333333333}{x} + \left(-3 + 3 \cdot y\right)\right)} \]

   6. Taylor expanded in y around inf 98.8%

      \[\leadsto \sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + \color{blue}{3 \cdot y}\right) \]

   if 7.79999999999999986e-4 < x

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 93.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-udef 93.3%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      3. *-commutative 93.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot 3}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      4. metadata-eval 93.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. sqrt-prod 93.3%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   3. Applied egg-rr 93.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   4. Step-by-step derivation

      1. expm1-def 93.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-log1p 99.7%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. Simplified 99.7%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   6. Taylor expanded in y around inf 98.9%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{y} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00078:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 8: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot \left(\left(\left(y + -1\right) + \frac{0.1111111111111111}{x}\right) \cdot 3\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (sqrt x) (* (+ (+ y -1.0) (/ 0.1111111111111111 x)) 3.0)))

C

double code(double x, double y) {
	return sqrt(x) * (((y + -1.0) + (0.1111111111111111 / x)) * 3.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x) * (((y + (-1.0d0)) + (0.1111111111111111d0 / x)) * 3.0d0)
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x) * (((y + -1.0) + (0.1111111111111111 / x)) * 3.0);
}

Python

def code(x, y):
	return math.sqrt(x) * (((y + -1.0) + (0.1111111111111111 / x)) * 3.0)

Julia

function code(x, y)
	return Float64(sqrt(x) * Float64(Float64(Float64(y + -1.0) + Float64(0.1111111111111111 / x)) * 3.0))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x) * (((y + -1.0) + (0.1111111111111111 / x)) * 3.0);
end

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[(y + -1.0), $MachinePrecision] + N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   2. associate-*l* 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

   3. sub-neg 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

   4. +-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right)\right) \]

   5. +-lft-identity 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(\frac{1}{x \cdot 9} + \color{blue}{\left(0 + y\right)}\right) + \left(-1\right)\right)\right) \]

   6. associate-+l+ 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)}\right) \]

   7. *-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

   8. associate-/r* 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

   9. metadata-eval 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

   10. +-lft-identity 99.4%

       \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(\color{blue}{y} + \left(-1\right)\right)\right)\right) \]

   11. metadata-eval 99.4%

       \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + \color{blue}{-1}\right)\right)\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]

4. Final simplification 99.4%

   \[\leadsto \sqrt{x} \cdot \left(\left(\left(y + -1\right) + \frac{0.1111111111111111}{x}\right) \cdot 3\right) \]

Alternative 9: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + \left(-3 + y \cdot 3\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (sqrt x) (+ (/ 0.3333333333333333 x) (+ -3.0 (* y 3.0)))))

C

double code(double x, double y) {
	return sqrt(x) * ((0.3333333333333333 / x) + (-3.0 + (y * 3.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x) * ((0.3333333333333333d0 / x) + ((-3.0d0) + (y * 3.0d0)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x) * ((0.3333333333333333 / x) + (-3.0 + (y * 3.0)));
}

Python

def code(x, y):
	return math.sqrt(x) * ((0.3333333333333333 / x) + (-3.0 + (y * 3.0)))

Julia

function code(x, y)
	return Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + Float64(-3.0 + Float64(y * 3.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x) * ((0.3333333333333333 / x) + (-3.0 + (y * 3.0)));
end

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + N[(-3.0 + N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   2. associate-*l* 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

   3. sub-neg 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

   4. +-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right)\right) \]

   5. +-lft-identity 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(\frac{1}{x \cdot 9} + \color{blue}{\left(0 + y\right)}\right) + \left(-1\right)\right)\right) \]

   6. associate-+l+ 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)}\right) \]

   7. *-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

   8. associate-/r* 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

   9. metadata-eval 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

   10. +-lft-identity 99.4%

       \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(\color{blue}{y} + \left(-1\right)\right)\right)\right) \]

   11. metadata-eval 99.4%

       \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + \color{blue}{-1}\right)\right)\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]
4. Step-by-step derivation

   1. distribute-lft-in 99.4%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \frac{0.1111111111111111}{x} + 3 \cdot \left(y + -1\right)\right)} \]

   2. associate-*r/ 99.4%

      \[\leadsto \sqrt{x} \cdot \left(\color{blue}{\frac{3 \cdot 0.1111111111111111}{x}} + 3 \cdot \left(y + -1\right)\right) \]

   3. metadata-eval 99.4%

      \[\leadsto \sqrt{x} \cdot \left(\frac{\color{blue}{0.3333333333333333}}{x} + 3 \cdot \left(y + -1\right)\right) \]

   4. +-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + 3 \cdot \color{blue}{\left(-1 + y\right)}\right) \]

   5. distribute-lft-in 99.5%

      \[\leadsto \sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + \color{blue}{\left(3 \cdot -1 + 3 \cdot y\right)}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + \left(\color{blue}{-3} + 3 \cdot y\right)\right) \]

5. Applied egg-rr 99.5%

   \[\leadsto \sqrt{x} \cdot \color{blue}{\left(\frac{0.3333333333333333}{x} + \left(-3 + 3 \cdot y\right)\right)} \]

6. Final simplification 99.5%

   \[\leadsto \sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + \left(-3 + y \cdot 3\right)\right) \]

Alternative 10: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot 9} \cdot \left(\left(y + -1\right) + \frac{0.1111111111111111}{x}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (sqrt (* x 9.0)) (+ (+ y -1.0) (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	return sqrt((x * 9.0)) * ((y + -1.0) + (0.1111111111111111 / x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((x * 9.0d0)) * ((y + (-1.0d0)) + (0.1111111111111111d0 / x))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt((x * 9.0)) * ((y + -1.0) + (0.1111111111111111 / x));
}

Python

def code(x, y):
	return math.sqrt((x * 9.0)) * ((y + -1.0) + (0.1111111111111111 / x))

Julia

function code(x, y)
	return Float64(sqrt(Float64(x * 9.0)) * Float64(Float64(y + -1.0) + Float64(0.1111111111111111 / x)))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((x * 9.0)) * ((y + -1.0) + (0.1111111111111111 / x));
end

Wolfram

code[x_, y_] := N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(N[(y + -1.0), $MachinePrecision] + N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   2. associate-*l* 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

   3. sub-neg 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

   4. +-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right)\right) \]

   5. +-lft-identity 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(\frac{1}{x \cdot 9} + \color{blue}{\left(0 + y\right)}\right) + \left(-1\right)\right)\right) \]

   6. associate-+l+ 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)}\right) \]

   7. *-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

   8. associate-/r* 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

   9. metadata-eval 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(\left(0 + y\right) + \left(-1\right)\right)\right)\right) \]

   10. +-lft-identity 99.4%

       \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(\color{blue}{y} + \left(-1\right)\right)\right)\right) \]

   11. metadata-eval 99.4%

       \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + \color{blue}{-1}\right)\right)\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]
4. Step-by-step derivation

   1. metadata-eval 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{\color{blue}{\frac{1}{9}}}{x} + \left(y + -1\right)\right)\right) \]

   2. associate-/r* 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\frac{1}{9 \cdot x}} + \left(y + -1\right)\right)\right) \]

   3. *-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\frac{1}{\color{blue}{x \cdot 9}} + \left(y + -1\right)\right)\right) \]

   4. associate-+l+ 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(\frac{1}{x \cdot 9} + y\right) + -1\right)}\right) \]

   5. +-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} + -1\right)\right) \]

   6. metadata-eval 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + \color{blue}{\left(-1\right)}\right)\right) \]

   7. sub-neg 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)}\right) \]

   8. associate-*l* 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)} \]

   9. *-commutative 99.4%

      \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   10. associate--l+ 99.4%

       \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)} \]

   11. distribute-lft-in 99.4%

       \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right)} \]

   12. *-commutative 99.4%

       \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

   13. metadata-eval 99.4%

       \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

   14. sqrt-prod 99.5%

       \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

   15. *-commutative 99.5%

       \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

   16. metadata-eval 99.5%

       \[\leadsto \sqrt{x \cdot 9} \cdot y + \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

   17. sqrt-prod 99.6%

       \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\frac{1}{x \cdot 9} - 1\right) \]

5. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot y + \sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)} \]
6. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \color{blue}{y \cdot \sqrt{x \cdot 9}} + \sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   2. distribute-lft-in 99.5%

      \[\leadsto y \cdot \sqrt{x \cdot 9} + \color{blue}{\left(\sqrt{x \cdot 9} \cdot \frac{0.1111111111111111}{x} + \sqrt{x \cdot 9} \cdot -1\right)} \]

   3. *-commutative 99.5%

      \[\leadsto y \cdot \sqrt{x \cdot 9} + \left(\color{blue}{\frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9}} + \sqrt{x \cdot 9} \cdot -1\right) \]

   4. +-commutative 99.5%

      \[\leadsto y \cdot \sqrt{x \cdot 9} + \color{blue}{\left(\sqrt{x \cdot 9} \cdot -1 + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9}\right)} \]

   5. associate-+l+ 99.5%

      \[\leadsto \color{blue}{\left(y \cdot \sqrt{x \cdot 9} + \sqrt{x \cdot 9} \cdot -1\right) + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9}} \]

   6. *-commutative 99.5%

      \[\leadsto \left(\color{blue}{\sqrt{x \cdot 9} \cdot y} + \sqrt{x \cdot 9} \cdot -1\right) + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9} \]

   7. distribute-lft-in 99.5%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(y + -1\right)} + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9} \]

   8. *-commutative 99.5%

      \[\leadsto \color{blue}{\left(y + -1\right) \cdot \sqrt{x \cdot 9}} + \frac{0.1111111111111111}{x} \cdot \sqrt{x \cdot 9} \]

   9. distribute-rgt-out 99.5%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(\left(y + -1\right) + \frac{0.1111111111111111}{x}\right)} \]

   10. +-commutative 99.5%

       \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)} \]

7. Simplified 99.5%

   \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)} \]

8. Final simplification 99.5%

   \[\leadsto \sqrt{x \cdot 9} \cdot \left(\left(y + -1\right) + \frac{0.1111111111111111}{x}\right) \]

Alternative 11: 61.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00078:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00078) (sqrt (/ 0.1111111111111111 x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00078) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.00078d0) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00078) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00078:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00078)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00078)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00078], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.79999999999999986e-4

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. expm1-log1p-u 99.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-udef 7.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      3. *-commutative 7.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot 3}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      4. metadata-eval 7.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. sqrt-prod 7.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   3. Applied egg-rr 7.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   4. Step-by-step derivation

      1. expm1-def 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-log1p 99.5%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. Simplified 99.5%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   6. Step-by-step derivation

      1. sub-neg 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)} \]

      2. +-commutative 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right) \]

      3. metadata-eval 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\left(\frac{1}{x \cdot 9} + y\right) + \color{blue}{-1}\right) \]

      4. associate-+l+ 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(y + -1\right)\right)} \]

      5. *-commutative 99.5%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(y + -1\right)\right) \]

      6. associate-/r* 99.3%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(y + -1\right)\right) \]

      7. metadata-eval 99.3%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(y + -1\right)\right) \]

      8. add-sqr-sqrt 85.9%

         \[\leadsto \color{blue}{\sqrt{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)} \cdot \sqrt{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}} \]

      9. sqrt-unprod 80.6%

         \[\leadsto \color{blue}{\sqrt{\left(\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right) \cdot \left(\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)}} \]

      10. swap-sqr 33.8%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x \cdot 9} \cdot \sqrt{x \cdot 9}\right) \cdot \left(\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)}} \]

      11. add-sqr-sqrt 33.8%

          \[\leadsto \sqrt{\color{blue}{\left(x \cdot 9\right)} \cdot \left(\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]

      12. pow2 33.8%

          \[\leadsto \sqrt{\left(x \cdot 9\right) \cdot \color{blue}{{\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}^{2}}} \]

   7. Applied egg-rr 33.8%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot 9\right) \cdot {\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}^{2}}} \]

   8. Taylor expanded in x around 0 75.4%

      \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \]

   if 7.79999999999999986e-4 < x

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative 99.5%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. associate-*l* 99.5%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      3. sub-neg 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

      4. +-commutative 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(-1\right) + \left(y + \frac{1}{x \cdot 9}\right)\right)}\right) \]

      5. distribute-lft-in 99.5%

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(-1\right) + 3 \cdot \left(y + \frac{1}{x \cdot 9}\right)\right)} \]

      6. metadata-eval 99.5%

         \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{-1} + 3 \cdot \left(y + \frac{1}{x \cdot 9}\right)\right) \]

      7. metadata-eval 99.5%

         \[\leadsto \sqrt{x} \cdot \left(\color{blue}{-3} + 3 \cdot \left(y + \frac{1}{x \cdot 9}\right)\right) \]

      8. distribute-lft-in 99.5%

         \[\leadsto \sqrt{x} \cdot \left(-3 + \color{blue}{\left(3 \cdot y + 3 \cdot \frac{1}{x \cdot 9}\right)}\right) \]

      9. *-commutative 99.5%

         \[\leadsto \sqrt{x} \cdot \left(-3 + \left(3 \cdot y + \color{blue}{\frac{1}{x \cdot 9} \cdot 3}\right)\right) \]

      10. fma-def 99.5%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \color{blue}{\mathsf{fma}\left(3, y, \frac{1}{x \cdot 9} \cdot 3\right)}\right) \]

      11. *-commutative 99.5%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{1}{\color{blue}{9 \cdot x}} \cdot 3\right)\right) \]

      12. associate-/r* 99.5%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \color{blue}{\frac{\frac{1}{9}}{x}} \cdot 3\right)\right) \]

      13. associate-*l/ 99.5%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \color{blue}{\frac{\frac{1}{9} \cdot 3}{x}}\right)\right) \]

      14. metadata-eval 99.5%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{\color{blue}{0.1111111111111111} \cdot 3}{x}\right)\right) \]

      15. metadata-eval 99.5%

          \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{\color{blue}{0.3333333333333333}}{x}\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\right)} \]

   4. Taylor expanded in x around inf 98.8%

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

   5. Taylor expanded in y around 0 54.8%

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

      1. *-commutative 54.8%

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

   7. Simplified 54.8%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00078:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 12: 3.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot 9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (* x 9.0)))

C

double code(double x, double y) {
	return sqrt((x * 9.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((x * 9.0d0))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt((x * 9.0));
}

Python

def code(x, y):
	return math.sqrt((x * 9.0))

Julia

function code(x, y)
	return sqrt(Float64(x * 9.0))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((x * 9.0));
end

Wolfram

code[x_, y_] := N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Step-by-step derivation

   1. *-commutative 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   2. associate-*l* 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

   3. sub-neg 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)}\right) \]

   4. +-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(\left(-1\right) + \left(y + \frac{1}{x \cdot 9}\right)\right)}\right) \]

   5. distribute-lft-in 99.4%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(-1\right) + 3 \cdot \left(y + \frac{1}{x \cdot 9}\right)\right)} \]

   6. metadata-eval 99.4%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{-1} + 3 \cdot \left(y + \frac{1}{x \cdot 9}\right)\right) \]

   7. metadata-eval 99.4%

      \[\leadsto \sqrt{x} \cdot \left(\color{blue}{-3} + 3 \cdot \left(y + \frac{1}{x \cdot 9}\right)\right) \]

   8. distribute-lft-in 99.4%

      \[\leadsto \sqrt{x} \cdot \left(-3 + \color{blue}{\left(3 \cdot y + 3 \cdot \frac{1}{x \cdot 9}\right)}\right) \]

   9. *-commutative 99.4%

      \[\leadsto \sqrt{x} \cdot \left(-3 + \left(3 \cdot y + \color{blue}{\frac{1}{x \cdot 9} \cdot 3}\right)\right) \]

   10. fma-def 99.4%

       \[\leadsto \sqrt{x} \cdot \left(-3 + \color{blue}{\mathsf{fma}\left(3, y, \frac{1}{x \cdot 9} \cdot 3\right)}\right) \]

   11. *-commutative 99.4%

       \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{1}{\color{blue}{9 \cdot x}} \cdot 3\right)\right) \]

   12. associate-/r* 99.4%

       \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \color{blue}{\frac{\frac{1}{9}}{x}} \cdot 3\right)\right) \]

   13. associate-*l/ 99.5%

       \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \color{blue}{\frac{\frac{1}{9} \cdot 3}{x}}\right)\right) \]

   14. metadata-eval 99.5%

       \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{\color{blue}{0.1111111111111111} \cdot 3}{x}\right)\right) \]

   15. metadata-eval 99.5%

       \[\leadsto \sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{\color{blue}{0.3333333333333333}}{x}\right)\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \left(-3 + \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\right)} \]

4. Taylor expanded in x around inf 61.2%

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

5. Taylor expanded in y around 0 27.9%

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

   1. *-commutative 27.9%

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

7. Simplified 27.9%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 3.0%

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

   3. swap-sqr 3.0%

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

   4. add-sqr-sqrt 3.0%

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

   5. metadata-eval 3.0%

      \[\leadsto \sqrt{x \cdot \color{blue}{9}} \]

   6. pow1/2 3.0%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \]

9. Applied egg-rr 3.0%

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

    1. unpow1/2 3.0%

       \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \]

11. Simplified 3.0%

    \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \]

12. Final simplification 3.0%

    \[\leadsto \sqrt{x \cdot 9} \]

Alternative 13: 38.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.1111111111111111}{x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (/ 0.1111111111111111 x)))

C

double code(double x, double y) {
	return sqrt((0.1111111111111111 / x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((0.1111111111111111d0 / x))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt((0.1111111111111111 / x));
}

Python

def code(x, y):
	return math.sqrt((0.1111111111111111 / x))

Julia

function code(x, y)
	return sqrt(Float64(0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((0.1111111111111111 / x));
end

Wolfram

code[x_, y_] := N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Step-by-step derivation

   1. expm1-log1p-u 96.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   2. expm1-udef 49.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   3. *-commutative 49.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot 3}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   4. metadata-eval 49.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   5. sqrt-prod 49.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

3. Applied egg-rr 49.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
4. Step-by-step derivation

   1. expm1-def 96.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   2. expm1-log1p 99.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
6. Step-by-step derivation

   1. sub-neg 99.6%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) + \left(-1\right)\right)} \]

   2. +-commutative 99.6%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\left(\frac{1}{x \cdot 9} + y\right)} + \left(-1\right)\right) \]

   3. metadata-eval 99.6%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\left(\frac{1}{x \cdot 9} + y\right) + \color{blue}{-1}\right) \]

   4. associate-+l+ 99.6%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(y + -1\right)\right)} \]

   5. *-commutative 99.6%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(y + -1\right)\right) \]

   6. associate-/r* 99.5%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(y + -1\right)\right) \]

   7. metadata-eval 99.5%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(y + -1\right)\right) \]

   8. add-sqr-sqrt 51.7%

      \[\leadsto \color{blue}{\sqrt{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)} \cdot \sqrt{\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}} \]

   9. sqrt-unprod 45.8%

      \[\leadsto \color{blue}{\sqrt{\left(\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right) \cdot \left(\sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)}} \]

   10. swap-sqr 22.0%

       \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x \cdot 9} \cdot \sqrt{x \cdot 9}\right) \cdot \left(\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)}} \]

   11. add-sqr-sqrt 22.1%

       \[\leadsto \sqrt{\color{blue}{\left(x \cdot 9\right)} \cdot \left(\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)\right)} \]

   12. pow2 22.1%

       \[\leadsto \sqrt{\left(x \cdot 9\right) \cdot \color{blue}{{\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}^{2}}} \]

7. Applied egg-rr 22.1%

   \[\leadsto \color{blue}{\sqrt{\left(x \cdot 9\right) \cdot {\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right)}^{2}}} \]

8. Taylor expanded in x around 0 39.1%

   \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \]

9. Final simplification 39.1%

   \[\leadsto \sqrt{\frac{0.1111111111111111}{x}} \]

Developer target: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x)))))

C

double code(double x, double y) {
	return 3.0 * ((y * sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 3.0d0 * ((y * sqrt(x)) + (((1.0d0 / (x * 9.0d0)) - 1.0d0) * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return 3.0 * ((y * Math.sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * Math.sqrt(x)));
}

Python

def code(x, y):
	return 3.0 * ((y * math.sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(3.0 * Float64(Float64(y * sqrt(x)) + Float64(Float64(Float64(1.0 / Float64(x * 9.0)) - 1.0) * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = 3.0 * ((y * sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * sqrt(x)));
end

Wolfram

code[x_, y_] := N[(3.0 * N[(N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023301 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))