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

Percentage Accurate: 99.4% → 99.4%

Time: 11.3s

Alternatives: 10

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(y + N[(N[(0.1111111111111111 / x), $MachinePrecision] + -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. associate--l+ 99.4%

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

   2. sub-neg 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. metadata-eval 99.4%

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

   3. sqrt-prod 99.6%

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

   4. pow1/2 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]

5. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]
6. Step-by-step derivation

   1. unpow1/2 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ t_1 := 3 \cdot \left(y \cdot \sqrt{x}\right)\\ t_2 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -0.0185:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-177}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-258}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-237}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 3000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x)))
        (t_1 (* 3.0 (* y (sqrt x))))
        (t_2 (* (sqrt x) -3.0)))
   (if (<= y -0.0185)
     t_1
     (if (<= y -2.8e-177)
       t_2
       (if (<= y -5.2e-258)
         t_0
         (if (<= y 4e-237) t_2 (if (<= y 3000000000.0) t_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 t_2 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -0.0185) {
		tmp = t_1;
	} else if (y <= -2.8e-177) {
		tmp = t_2;
	} else if (y <= -5.2e-258) {
		tmp = t_0;
	} else if (y <= 4e-237) {
		tmp = t_2;
	} else if (y <= 3000000000.0) {
		tmp = t_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) :: t_2
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    t_1 = 3.0d0 * (y * sqrt(x))
    t_2 = sqrt(x) * (-3.0d0)
    if (y <= (-0.0185d0)) then
        tmp = t_1
    else if (y <= (-2.8d-177)) then
        tmp = t_2
    else if (y <= (-5.2d-258)) then
        tmp = t_0
    else if (y <= 4d-237) then
        tmp = t_2
    else if (y <= 3000000000.0d0) then
        tmp = t_0
    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 t_2 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -0.0185) {
		tmp = t_1;
	} else if (y <= -2.8e-177) {
		tmp = t_2;
	} else if (y <= -5.2e-258) {
		tmp = t_0;
	} else if (y <= 4e-237) {
		tmp = t_2;
	} else if (y <= 3000000000.0) {
		tmp = t_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))
	t_2 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -0.0185:
		tmp = t_1
	elif y <= -2.8e-177:
		tmp = t_2
	elif y <= -5.2e-258:
		tmp = t_0
	elif y <= 4e-237:
		tmp = t_2
	elif y <= 3000000000.0:
		tmp = t_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)))
	t_2 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -0.0185)
		tmp = t_1;
	elseif (y <= -2.8e-177)
		tmp = t_2;
	elseif (y <= -5.2e-258)
		tmp = t_0;
	elseif (y <= 4e-237)
		tmp = t_2;
	elseif (y <= 3000000000.0)
		tmp = t_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));
	t_2 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -0.0185)
		tmp = t_1;
	elseif (y <= -2.8e-177)
		tmp = t_2;
	elseif (y <= -5.2e-258)
		tmp = t_0;
	elseif (y <= 4e-237)
		tmp = t_2;
	elseif (y <= 3000000000.0)
		tmp = t_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]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -0.0185], t$95$1, If[LessEqual[y, -2.8e-177], t$95$2, If[LessEqual[y, -5.2e-258], t$95$0, If[LessEqual[y, 4e-237], t$95$2, If[LessEqual[y, 3000000000.0], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0184999999999999991 or 3e9 < y

   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. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 79.7%

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

      1. *-commutative 79.7%

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

   6. Simplified 79.7%

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

   if -0.0184999999999999991 < y < -2.79999999999999987e-177 or -5.20000000000000036e-258 < y < 4e-237

   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* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate--l+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-lft-in 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. fma-def 99.6%

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

      10. metadata-eval 99.6%

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

      11. metadata-eval 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r* 99.4%

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

      14. associate-*r/ 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 64.3%

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

   5. Taylor expanded in y around 0 62.2%

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

      1. *-commutative 62.2%

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

   7. Simplified 62.2%

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

   if -2.79999999999999987e-177 < y < -5.20000000000000036e-258 or 4e-237 < y < 3e9

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 60.2%

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

      1. associate-*r* 60.3%

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

      2. expm1-log1p-u 56.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      3. expm1-udef 56.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]

      4. associate-*r/ 56.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(3 \cdot \sqrt{x}\right) \cdot 0.1111111111111111}{x}}\right)} - 1 \]

      5. *-commutative 56.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot 0.1111111111111111}{x}\right)} - 1 \]

      6. associate-*l* 56.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \left(3 \cdot 0.1111111111111111\right)}}{x}\right)} - 1 \]

      7. metadata-eval 56.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot \color{blue}{0.3333333333333333}}{x}\right)} - 1 \]

   6. Applied egg-rr 56.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 56.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)\right)} \]

      2. expm1-log1p 60.1%

         \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 0.3333333333333333}{x}} \]

      3. rem-square-sqrt 60.1%

         \[\leadsto \frac{\sqrt{x} \cdot 0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      4. associate-/l/ 60.2%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{x} \cdot 0.3333333333333333}{\sqrt{x}}}{\sqrt{x}}} \]

      5. *-commutative 60.2%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333 \cdot \sqrt{x}}}{\sqrt{x}}}{\sqrt{x}} \]

      6. associate-/l* 60.3%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{\frac{\sqrt{x}}{\sqrt{x}}}}}{\sqrt{x}} \]

      7. *-inverses 60.3%

         \[\leadsto \frac{\frac{0.3333333333333333}{\color{blue}{1}}}{\sqrt{x}} \]

      8. metadata-eval 60.3%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{\sqrt{x}} \]

   8. Simplified 60.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
   9. Step-by-step derivation

      1. metadata-eval 60.3%

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

      2. sqrt-div 60.6%

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

      3. pow1/2 60.6%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   10. Applied egg-rr 60.6%

       \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   11. Step-by-step derivation

       1. unpow1/2 60.6%

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

   12. Simplified 60.6%

       \[\leadsto \color{blue}{\sqrt{\frac{0.1111111111111111}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0185:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-258}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 3000000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \]

Alternative 3: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ t_1 := \sqrt{x \cdot 9} \cdot y\\ t_2 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -0.0185:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-177}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-257}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-237}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 430000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x)))
        (t_1 (* (sqrt (* x 9.0)) y))
        (t_2 (* (sqrt x) -3.0)))
   (if (<= y -0.0185)
     t_1
     (if (<= y -5.6e-177)
       t_2
       (if (<= y -6.4e-257)
         t_0
         (if (<= y 4.2e-237) t_2 (if (<= y 430000000000.0) t_0 t_1)))))))

C

double code(double x, double y) {
	double t_0 = sqrt((0.1111111111111111 / x));
	double t_1 = sqrt((x * 9.0)) * y;
	double t_2 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -0.0185) {
		tmp = t_1;
	} else if (y <= -5.6e-177) {
		tmp = t_2;
	} else if (y <= -6.4e-257) {
		tmp = t_0;
	} else if (y <= 4.2e-237) {
		tmp = t_2;
	} else if (y <= 430000000000.0) {
		tmp = t_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) :: t_2
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    t_1 = sqrt((x * 9.0d0)) * y
    t_2 = sqrt(x) * (-3.0d0)
    if (y <= (-0.0185d0)) then
        tmp = t_1
    else if (y <= (-5.6d-177)) then
        tmp = t_2
    else if (y <= (-6.4d-257)) then
        tmp = t_0
    else if (y <= 4.2d-237) then
        tmp = t_2
    else if (y <= 430000000000.0d0) then
        tmp = t_0
    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 = Math.sqrt((x * 9.0)) * y;
	double t_2 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -0.0185) {
		tmp = t_1;
	} else if (y <= -5.6e-177) {
		tmp = t_2;
	} else if (y <= -6.4e-257) {
		tmp = t_0;
	} else if (y <= 4.2e-237) {
		tmp = t_2;
	} else if (y <= 430000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((0.1111111111111111 / x))
	t_1 = math.sqrt((x * 9.0)) * y
	t_2 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -0.0185:
		tmp = t_1
	elif y <= -5.6e-177:
		tmp = t_2
	elif y <= -6.4e-257:
		tmp = t_0
	elif y <= 4.2e-237:
		tmp = t_2
	elif y <= 430000000000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(0.1111111111111111 / x))
	t_1 = Float64(sqrt(Float64(x * 9.0)) * y)
	t_2 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -0.0185)
		tmp = t_1;
	elseif (y <= -5.6e-177)
		tmp = t_2;
	elseif (y <= -6.4e-257)
		tmp = t_0;
	elseif (y <= 4.2e-237)
		tmp = t_2;
	elseif (y <= 430000000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((0.1111111111111111 / x));
	t_1 = sqrt((x * 9.0)) * y;
	t_2 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -0.0185)
		tmp = t_1;
	elseif (y <= -5.6e-177)
		tmp = t_2;
	elseif (y <= -6.4e-257)
		tmp = t_0;
	elseif (y <= 4.2e-237)
		tmp = t_2;
	elseif (y <= 430000000000.0)
		tmp = t_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[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -0.0185], t$95$1, If[LessEqual[y, -5.6e-177], t$95$2, If[LessEqual[y, -6.4e-257], t$95$0, If[LessEqual[y, 4.2e-237], t$95$2, If[LessEqual[y, 430000000000.0], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0184999999999999991 or 4.3e11 < y

   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. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 79.8%

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

   if -0.0184999999999999991 < y < -5.59999999999999973e-177 or -6.39999999999999971e-257 < y < 4.2000000000000002e-237

   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* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate--l+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-lft-in 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. fma-def 99.6%

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

      10. metadata-eval 99.6%

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

      11. metadata-eval 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r* 99.4%

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

      14. associate-*r/ 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 64.3%

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

   5. Taylor expanded in y around 0 62.2%

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

      1. *-commutative 62.2%

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

   7. Simplified 62.2%

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

   if -5.59999999999999973e-177 < y < -6.39999999999999971e-257 or 4.2000000000000002e-237 < y < 4.3e11

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 60.2%

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

      1. associate-*r* 60.3%

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

      2. expm1-log1p-u 56.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      3. expm1-udef 56.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]

      4. associate-*r/ 56.5%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(3 \cdot \sqrt{x}\right) \cdot 0.1111111111111111}{x}}\right)} - 1 \]

      5. *-commutative 56.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot 0.1111111111111111}{x}\right)} - 1 \]

      6. associate-*l* 56.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \left(3 \cdot 0.1111111111111111\right)}}{x}\right)} - 1 \]

      7. metadata-eval 56.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot \color{blue}{0.3333333333333333}}{x}\right)} - 1 \]

   6. Applied egg-rr 56.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 56.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)\right)} \]

      2. expm1-log1p 60.1%

         \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 0.3333333333333333}{x}} \]

      3. rem-square-sqrt 60.1%

         \[\leadsto \frac{\sqrt{x} \cdot 0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      4. associate-/l/ 60.2%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{x} \cdot 0.3333333333333333}{\sqrt{x}}}{\sqrt{x}}} \]

      5. *-commutative 60.2%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333 \cdot \sqrt{x}}}{\sqrt{x}}}{\sqrt{x}} \]

      6. associate-/l* 60.3%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{\frac{\sqrt{x}}{\sqrt{x}}}}}{\sqrt{x}} \]

      7. *-inverses 60.3%

         \[\leadsto \frac{\frac{0.3333333333333333}{\color{blue}{1}}}{\sqrt{x}} \]

      8. metadata-eval 60.3%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{\sqrt{x}} \]

   8. Simplified 60.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
   9. Step-by-step derivation

      1. metadata-eval 60.3%

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

      2. sqrt-div 60.6%

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

      3. pow1/2 60.6%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   10. Applied egg-rr 60.6%

       \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   11. Step-by-step derivation

       1. unpow1/2 60.6%

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

   12. Simplified 60.6%

       \[\leadsto \color{blue}{\sqrt{\frac{0.1111111111111111}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0185:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-257}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 430000000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \]

Alternative 4: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot 9} \cdot y\\ t_1 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -0.0185:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-271}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{x}{0.3333333333333333}}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-237}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1550000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt (* x 9.0)) y)) (t_1 (* (sqrt x) -3.0)))
   (if (<= y -0.0185)
     t_0
     (if (<= y -1.25e-177)
       t_1
       (if (<= y -5e-271)
         (/ (sqrt x) (/ x 0.3333333333333333))
         (if (<= y 4.3e-237)
           t_1
           (if (<= y 1550000000.0) (sqrt (/ 0.1111111111111111 x)) t_0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt((x * 9.0)) * y;
	double t_1 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -0.0185) {
		tmp = t_0;
	} else if (y <= -1.25e-177) {
		tmp = t_1;
	} else if (y <= -5e-271) {
		tmp = sqrt(x) / (x / 0.3333333333333333);
	} else if (y <= 4.3e-237) {
		tmp = t_1;
	} else if (y <= 1550000000.0) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = t_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) :: t_1
    real(8) :: tmp
    t_0 = sqrt((x * 9.0d0)) * y
    t_1 = sqrt(x) * (-3.0d0)
    if (y <= (-0.0185d0)) then
        tmp = t_0
    else if (y <= (-1.25d-177)) then
        tmp = t_1
    else if (y <= (-5d-271)) then
        tmp = sqrt(x) / (x / 0.3333333333333333d0)
    else if (y <= 4.3d-237) then
        tmp = t_1
    else if (y <= 1550000000.0d0) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((x * 9.0)) * y;
	double t_1 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -0.0185) {
		tmp = t_0;
	} else if (y <= -1.25e-177) {
		tmp = t_1;
	} else if (y <= -5e-271) {
		tmp = Math.sqrt(x) / (x / 0.3333333333333333);
	} else if (y <= 4.3e-237) {
		tmp = t_1;
	} else if (y <= 1550000000.0) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((x * 9.0)) * y
	t_1 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -0.0185:
		tmp = t_0
	elif y <= -1.25e-177:
		tmp = t_1
	elif y <= -5e-271:
		tmp = math.sqrt(x) / (x / 0.3333333333333333)
	elif y <= 4.3e-237:
		tmp = t_1
	elif y <= 1550000000.0:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(Float64(x * 9.0)) * y)
	t_1 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -0.0185)
		tmp = t_0;
	elseif (y <= -1.25e-177)
		tmp = t_1;
	elseif (y <= -5e-271)
		tmp = Float64(sqrt(x) / Float64(x / 0.3333333333333333));
	elseif (y <= 4.3e-237)
		tmp = t_1;
	elseif (y <= 1550000000.0)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((x * 9.0)) * y;
	t_1 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -0.0185)
		tmp = t_0;
	elseif (y <= -1.25e-177)
		tmp = t_1;
	elseif (y <= -5e-271)
		tmp = sqrt(x) / (x / 0.3333333333333333);
	elseif (y <= 4.3e-237)
		tmp = t_1;
	elseif (y <= 1550000000.0)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -0.0185], t$95$0, If[LessEqual[y, -1.25e-177], t$95$1, If[LessEqual[y, -5e-271], N[(N[Sqrt[x], $MachinePrecision] / N[(x / 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e-237], t$95$1, If[LessEqual[y, 1550000000.0], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -0.0184999999999999991 or 1.55e9 < y

   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. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 79.8%

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

   if -0.0184999999999999991 < y < -1.25e-177 or -5.0000000000000002e-271 < y < 4.29999999999999982e-237

   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* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate--l+ 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-lft-in 99.6%

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

      7. sub-neg 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. fma-def 99.6%

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

      10. metadata-eval 99.6%

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

      11. metadata-eval 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r* 99.4%

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

      14. associate-*r/ 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 64.3%

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

   5. Taylor expanded in y around 0 62.2%

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

      1. *-commutative 62.2%

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

   7. Simplified 62.2%

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

   if -1.25e-177 < y < -5.0000000000000002e-271

   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. associate-*l* 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 74.1%

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

      1. associate-*r* 74.0%

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

      2. expm1-log1p-u 68.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      3. expm1-udef 69.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]

      4. associate-*r/ 69.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(3 \cdot \sqrt{x}\right) \cdot 0.1111111111111111}{x}}\right)} - 1 \]

      5. *-commutative 69.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot 0.1111111111111111}{x}\right)} - 1 \]

      6. associate-*l* 69.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \left(3 \cdot 0.1111111111111111\right)}}{x}\right)} - 1 \]

      7. metadata-eval 69.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot \color{blue}{0.3333333333333333}}{x}\right)} - 1 \]

   6. Applied egg-rr 69.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 68.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)\right)} \]

      2. expm1-log1p 73.9%

         \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 0.3333333333333333}{x}} \]

      3. associate-/l* 74.5%

         \[\leadsto \color{blue}{\frac{\sqrt{x}}{\frac{x}{0.3333333333333333}}} \]

   8. Simplified 74.5%

      \[\leadsto \color{blue}{\frac{\sqrt{x}}{\frac{x}{0.3333333333333333}}} \]

   if 4.29999999999999982e-237 < y < 1.55e9

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 56.8%

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

      1. associate-*r* 56.9%

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

      2. expm1-log1p-u 53.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      3. expm1-udef 53.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]

      4. associate-*r/ 53.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(3 \cdot \sqrt{x}\right) \cdot 0.1111111111111111}{x}}\right)} - 1 \]

      5. *-commutative 53.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot 0.1111111111111111}{x}\right)} - 1 \]

      6. associate-*l* 53.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \left(3 \cdot 0.1111111111111111\right)}}{x}\right)} - 1 \]

      7. metadata-eval 53.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot \color{blue}{0.3333333333333333}}{x}\right)} - 1 \]

   6. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 53.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)\right)} \]

      2. expm1-log1p 56.8%

         \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 0.3333333333333333}{x}} \]

      3. rem-square-sqrt 56.7%

         \[\leadsto \frac{\sqrt{x} \cdot 0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      4. associate-/l/ 56.8%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{x} \cdot 0.3333333333333333}{\sqrt{x}}}{\sqrt{x}}} \]

      5. *-commutative 56.8%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333 \cdot \sqrt{x}}}{\sqrt{x}}}{\sqrt{x}} \]

      6. associate-/l* 56.8%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{\frac{\sqrt{x}}{\sqrt{x}}}}}{\sqrt{x}} \]

      7. *-inverses 56.8%

         \[\leadsto \frac{\frac{0.3333333333333333}{\color{blue}{1}}}{\sqrt{x}} \]

      8. metadata-eval 56.8%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{\sqrt{x}} \]

   8. Simplified 56.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
   9. Step-by-step derivation

      1. metadata-eval 56.8%

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

      2. sqrt-div 57.2%

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

      3. pow1/2 57.2%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   10. Applied egg-rr 57.2%

       \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   11. Step-by-step derivation

       1. unpow1/2 57.2%

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

   12. Simplified 57.2%

       \[\leadsto \color{blue}{\sqrt{\frac{0.1111111111111111}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0185:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-177}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-271}:\\ \;\;\;\;\frac{\sqrt{x}}{\frac{x}{0.3333333333333333}}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-237}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1550000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \]

Alternative 5: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9500 \lor \neg \left(y \leq 1200000000\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9500.0) (not (<= y 1200000000.0)))
   (* (sqrt (* x 9.0)) y)
   (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9500.0) || !(y <= 1200000000.0)) {
		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 <= (-9500.0d0)) .or. (.not. (y <= 1200000000.0d0))) 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 <= -9500.0) || !(y <= 1200000000.0)) {
		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 <= -9500.0) or not (y <= 1200000000.0):
		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 <= -9500.0) || !(y <= 1200000000.0))
		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 <= -9500.0) || ~((y <= 1200000000.0)))
		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, -9500.0], N[Not[LessEqual[y, 1200000000.0]], $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 < -9500 or 1.2e9 < y

   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. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 81.0%

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

   if -9500 < y < 1.2e9

   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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. sub-neg 99.4%

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

      8. distribute-lft-in 99.5%

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

      9. fma-def 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.5%

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

      14. associate-*r/ 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 96.5%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{x} - 3\right) \cdot \sqrt{x}} \]
   5. Step-by-step derivation

      1. *-commutative 96.5%

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

      2. sub-neg 96.5%

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

      3. associate-*r/ 96.5%

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

      4. metadata-eval 96.5%

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

      5. metadata-eval 96.5%

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

   6. Simplified 96.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9500 \lor \neg \left(y \leq 1200000000\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 6: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \lor \neg \left(y \leq 2.25 \cdot 10^{-12}\right):\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.0) (not (<= y 2.25e-12)))
   (* (sqrt x) (- (* y 3.0) 3.0))
   (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.0) || !(y <= 2.25e-12)) {
		tmp = sqrt(x) * ((y * 3.0) - 3.0);
	} 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 <= (-2.0d0)) .or. (.not. (y <= 2.25d-12))) then
        tmp = sqrt(x) * ((y * 3.0d0) - 3.0d0)
    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 <= -2.0) || !(y <= 2.25e-12)) {
		tmp = Math.sqrt(x) * ((y * 3.0) - 3.0);
	} else {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.0) or not (y <= 2.25e-12):
		tmp = math.sqrt(x) * ((y * 3.0) - 3.0)
	else:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.0) || !(y <= 2.25e-12))
		tmp = Float64(sqrt(x) * Float64(Float64(y * 3.0) - 3.0));
	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 <= -2.0) || ~((y <= 2.25e-12)))
		tmp = sqrt(x) * ((y * 3.0) - 3.0);
	else
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.0], N[Not[LessEqual[y, 2.25e-12]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(y * 3.0), $MachinePrecision] - 3.0), $MachinePrecision]), $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 or 2.2499999999999999e-12 < y

   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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.3%

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

      7. sub-neg 99.3%

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

      8. distribute-lft-in 99.4%

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

      9. fma-def 99.4%

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

      10. metadata-eval 99.4%

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

      11. metadata-eval 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r* 99.4%

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

      14. associate-*r/ 99.5%

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

      15. metadata-eval 99.5%

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

      16. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 81.0%

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

   if -2 < y < 2.2499999999999999e-12

   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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. sub-neg 99.4%

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

      8. distribute-lft-in 99.5%

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

      9. fma-def 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.4%

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

      14. associate-*r/ 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 98.4%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{x} - 3\right) \cdot \sqrt{x}} \]
   5. Step-by-step derivation

      1. *-commutative 98.4%

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

      2. sub-neg 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

      5. metadata-eval 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \lor \neg \left(y \leq 2.25 \cdot 10^{-12}\right):\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 7: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -210 \lor \neg \left(y \leq 2.25 \cdot 10^{-12}\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -210.0) (not (<= y 2.25e-12)))
   (* (sqrt (* x 9.0)) (+ y -1.0))
   (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -210.0) || !(y <= 2.25e-12)) {
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	} 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 <= (-210.0d0)) .or. (.not. (y <= 2.25d-12))) then
        tmp = sqrt((x * 9.0d0)) * (y + (-1.0d0))
    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 <= -210.0) || !(y <= 2.25e-12)) {
		tmp = Math.sqrt((x * 9.0)) * (y + -1.0);
	} else {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -210.0) or not (y <= 2.25e-12):
		tmp = math.sqrt((x * 9.0)) * (y + -1.0)
	else:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -210.0) || !(y <= 2.25e-12))
		tmp = Float64(sqrt(Float64(x * 9.0)) * Float64(y + -1.0));
	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 <= -210.0) || ~((y <= 2.25e-12)))
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	else
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -210.0], N[Not[LessEqual[y, 2.25e-12]], $MachinePrecision]], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $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 < -210 or 2.2499999999999999e-12 < y

   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. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \]
   6. Step-by-step derivation

      1. unpow1/2 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 81.2%

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

   if -210 < y < 2.2499999999999999e-12

   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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. sub-neg 99.4%

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

      8. distribute-lft-in 99.5%

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

      9. fma-def 99.5%

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

      10. metadata-eval 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r* 99.4%

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

      14. associate-*r/ 99.4%

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

      15. metadata-eval 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 98.4%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{x} - 3\right) \cdot \sqrt{x}} \]
   5. Step-by-step derivation

      1. *-commutative 98.4%

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

      2. sub-neg 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

      5. metadata-eval 98.4%

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

   6. Simplified 98.4%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -210 \lor \neg \left(y \leq 2.25 \cdot 10^{-12}\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]

Alternative 8: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return 3.0 * ((y + ((0.1111111111111111 / x) + -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 + ((0.1111111111111111d0 / x) + (-1.0d0))) * sqrt(x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(3.0 * N[(N[(y + N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[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. associate-*l* 99.4%

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

   2. associate--l+ 99.4%

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

   3. sub-neg 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 9: 61.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 240000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 240000000.0) (sqrt (/ 0.1111111111111111 x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 240000000.0) {
		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 <= 240000000.0d0) 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 <= 240000000.0) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 240000000.0:
		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 <= 240000000.0)
		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 <= 240000000.0)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 240000000.0], 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 < 2.4e8

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 65.3%

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

      1. associate-*r* 65.4%

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

      2. expm1-log1p-u 61.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

      3. expm1-udef 61.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]

      4. associate-*r/ 61.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(3 \cdot \sqrt{x}\right) \cdot 0.1111111111111111}{x}}\right)} - 1 \]

      5. *-commutative 61.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot 0.1111111111111111}{x}\right)} - 1 \]

      6. associate-*l* 61.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \left(3 \cdot 0.1111111111111111\right)}}{x}\right)} - 1 \]

      7. metadata-eval 61.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot \color{blue}{0.3333333333333333}}{x}\right)} - 1 \]

   6. Applied egg-rr 61.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 61.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)\right)} \]

      2. expm1-log1p 65.3%

         \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 0.3333333333333333}{x}} \]

      3. rem-square-sqrt 65.3%

         \[\leadsto \frac{\sqrt{x} \cdot 0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

      4. associate-/l/ 65.4%

         \[\leadsto \color{blue}{\frac{\frac{\sqrt{x} \cdot 0.3333333333333333}{\sqrt{x}}}{\sqrt{x}}} \]

      5. *-commutative 65.4%

         \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333 \cdot \sqrt{x}}}{\sqrt{x}}}{\sqrt{x}} \]

      6. associate-/l* 65.4%

         \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{\frac{\sqrt{x}}{\sqrt{x}}}}}{\sqrt{x}} \]

      7. *-inverses 65.4%

         \[\leadsto \frac{\frac{0.3333333333333333}{\color{blue}{1}}}{\sqrt{x}} \]

      8. metadata-eval 65.4%

         \[\leadsto \frac{\color{blue}{0.3333333333333333}}{\sqrt{x}} \]

   8. Simplified 65.4%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
   9. Step-by-step derivation

      1. metadata-eval 65.4%

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

      2. sqrt-div 65.7%

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

      3. pow1/2 65.7%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   10. Applied egg-rr 65.7%

       \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   11. Step-by-step derivation

       1. unpow1/2 65.7%

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

   12. Simplified 65.7%

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

   if 2.4e8 < 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. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-lft-in 99.5%

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

      7. sub-neg 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. fma-def 99.6%

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

      10. metadata-eval 99.6%

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

      11. metadata-eval 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r* 99.6%

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

      14. associate-*r/ 99.6%

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

      15. metadata-eval 99.6%

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 99.4%

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

   5. Taylor expanded in y around 0 46.5%

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

      1. *-commutative 46.5%

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

   7. Simplified 46.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 240000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 10: 38.1% 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. associate-*l* 99.4%

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

   2. associate--l+ 99.4%

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

   3. sub-neg 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in x around 0 33.8%

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

   1. associate-*r* 33.9%

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

   2. expm1-log1p-u 31.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)\right)} \]

   3. expm1-udef 31.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}\right)} - 1} \]

   4. associate-*r/ 31.9%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\left(3 \cdot \sqrt{x}\right) \cdot 0.1111111111111111}{x}}\right)} - 1 \]

   5. *-commutative 31.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot 0.1111111111111111}{x}\right)} - 1 \]

   6. associate-*l* 31.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \left(3 \cdot 0.1111111111111111\right)}}{x}\right)} - 1 \]

   7. metadata-eval 31.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot \color{blue}{0.3333333333333333}}{x}\right)} - 1 \]

6. Applied egg-rr 31.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)} - 1} \]
7. Step-by-step derivation

   1. expm1-def 31.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{x} \cdot 0.3333333333333333}{x}\right)\right)} \]

   2. expm1-log1p 33.8%

      \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot 0.3333333333333333}{x}} \]

   3. rem-square-sqrt 33.8%

      \[\leadsto \frac{\sqrt{x} \cdot 0.3333333333333333}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

   4. associate-/l/ 33.9%

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{x} \cdot 0.3333333333333333}{\sqrt{x}}}{\sqrt{x}}} \]

   5. *-commutative 33.9%

      \[\leadsto \frac{\frac{\color{blue}{0.3333333333333333 \cdot \sqrt{x}}}{\sqrt{x}}}{\sqrt{x}} \]

   6. associate-/l* 33.9%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{\frac{\sqrt{x}}{\sqrt{x}}}}}{\sqrt{x}} \]

   7. *-inverses 33.9%

      \[\leadsto \frac{\frac{0.3333333333333333}{\color{blue}{1}}}{\sqrt{x}} \]

   8. metadata-eval 33.9%

      \[\leadsto \frac{\color{blue}{0.3333333333333333}}{\sqrt{x}} \]

8. Simplified 33.9%

   \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
9. Step-by-step derivation

   1. metadata-eval 33.9%

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

   2. sqrt-div 34.0%

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

   3. pow1/2 34.0%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

10. Applied egg-rr 34.0%

    \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
11. Step-by-step derivation

    1. unpow1/2 34.0%

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

12. Simplified 34.0%

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

13. Final simplification 34.0%

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

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 2023279 
(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)))