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

Percentage Accurate: 99.4% → 99.4%

Time: 9.6s

Alternatives: 11

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%

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

   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. Step-by-step derivation

   1. associate-*r* 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-commutative 99.3%

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

   4. metadata-eval 99.3%

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   5. sqrt-prod 99.4%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   6. *-commutative 99.4%

      \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   7. metadata-eval 99.4%

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

   8. sqrt-prod 99.5%

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

5. Applied egg-rr 99.5%

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

   1. distribute-lft-out 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 62.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ t_1 := 3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{if}\;x \leq 1.85 \cdot 10^{-166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+14} \lor \neg \left(x \leq 5.5 \cdot 10^{+24}\right) \land \left(x \leq 3.8 \cdot 10^{+39} \lor \neg \left(x \leq 7 \cdot 10^{+54}\right) \land \left(x \leq 2 \cdot 10^{+77} \lor \neg \left(x \leq 1.55 \cdot 10^{+111}\right) \land x \leq 1.3 \cdot 10^{+139}\right)\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x))) (t_1 (* 3.0 (* y (sqrt x)))))
   (if (<= x 1.85e-166)
     t_0
     (if (<= x 3.6e-163)
       t_1
       (if (<= x 1.18e-51)
         t_0
         (if (or (<= x 4.3e+14)
                 (and (not (<= x 5.5e+24))
                      (or (<= x 3.8e+39)
                          (and (not (<= x 7e+54))
                               (or (<= x 2e+77)
                                   (and (not (<= x 1.55e+111))
                                        (<= x 1.3e+139)))))))
           t_1
           (* (sqrt x) -3.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt((0.1111111111111111 / x));
	double t_1 = 3.0 * (y * sqrt(x));
	double tmp;
	if (x <= 1.85e-166) {
		tmp = t_0;
	} else if (x <= 3.6e-163) {
		tmp = t_1;
	} else if (x <= 1.18e-51) {
		tmp = t_0;
	} else if ((x <= 4.3e+14) || (!(x <= 5.5e+24) && ((x <= 3.8e+39) || (!(x <= 7e+54) && ((x <= 2e+77) || (!(x <= 1.55e+111) && (x <= 1.3e+139))))))) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    t_1 = 3.0d0 * (y * sqrt(x))
    if (x <= 1.85d-166) then
        tmp = t_0
    else if (x <= 3.6d-163) then
        tmp = t_1
    else if (x <= 1.18d-51) then
        tmp = t_0
    else if ((x <= 4.3d+14) .or. (.not. (x <= 5.5d+24)) .and. (x <= 3.8d+39) .or. (.not. (x <= 7d+54)) .and. (x <= 2d+77) .or. (.not. (x <= 1.55d+111)) .and. (x <= 1.3d+139)) then
        tmp = t_1
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((0.1111111111111111 / x));
	double t_1 = 3.0 * (y * Math.sqrt(x));
	double tmp;
	if (x <= 1.85e-166) {
		tmp = t_0;
	} else if (x <= 3.6e-163) {
		tmp = t_1;
	} else if (x <= 1.18e-51) {
		tmp = t_0;
	} else if ((x <= 4.3e+14) || (!(x <= 5.5e+24) && ((x <= 3.8e+39) || (!(x <= 7e+54) && ((x <= 2e+77) || (!(x <= 1.55e+111) && (x <= 1.3e+139))))))) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((0.1111111111111111 / x))
	t_1 = 3.0 * (y * math.sqrt(x))
	tmp = 0
	if x <= 1.85e-166:
		tmp = t_0
	elif x <= 3.6e-163:
		tmp = t_1
	elif x <= 1.18e-51:
		tmp = t_0
	elif (x <= 4.3e+14) or (not (x <= 5.5e+24) and ((x <= 3.8e+39) or (not (x <= 7e+54) and ((x <= 2e+77) or (not (x <= 1.55e+111) and (x <= 1.3e+139)))))):
		tmp = t_1
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(0.1111111111111111 / x))
	t_1 = Float64(3.0 * Float64(y * sqrt(x)))
	tmp = 0.0
	if (x <= 1.85e-166)
		tmp = t_0;
	elseif (x <= 3.6e-163)
		tmp = t_1;
	elseif (x <= 1.18e-51)
		tmp = t_0;
	elseif ((x <= 4.3e+14) || (!(x <= 5.5e+24) && ((x <= 3.8e+39) || (!(x <= 7e+54) && ((x <= 2e+77) || (!(x <= 1.55e+111) && (x <= 1.3e+139)))))))
		tmp = t_1;
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((0.1111111111111111 / x));
	t_1 = 3.0 * (y * sqrt(x));
	tmp = 0.0;
	if (x <= 1.85e-166)
		tmp = t_0;
	elseif (x <= 3.6e-163)
		tmp = t_1;
	elseif (x <= 1.18e-51)
		tmp = t_0;
	elseif ((x <= 4.3e+14) || (~((x <= 5.5e+24)) && ((x <= 3.8e+39) || (~((x <= 7e+54)) && ((x <= 2e+77) || (~((x <= 1.55e+111)) && (x <= 1.3e+139)))))))
		tmp = t_1;
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.85e-166], t$95$0, If[LessEqual[x, 3.6e-163], t$95$1, If[LessEqual[x, 1.18e-51], t$95$0, If[Or[LessEqual[x, 4.3e+14], And[N[Not[LessEqual[x, 5.5e+24]], $MachinePrecision], Or[LessEqual[x, 3.8e+39], And[N[Not[LessEqual[x, 7e+54]], $MachinePrecision], Or[LessEqual[x, 2e+77], And[N[Not[LessEqual[x, 1.55e+111]], $MachinePrecision], LessEqual[x, 1.3e+139]]]]]]], t$95$1, N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.8500000000000001e-166 or 3.5999999999999998e-163 < x < 1.18000000000000004e-51

   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* 99.5%

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

      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* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Applied egg-rr 28.4%

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

      1. associate-*r* 28.4%

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

      2. *-commutative 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in x around 0 83.7%

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

   if 1.8500000000000001e-166 < x < 3.5999999999999998e-163 or 1.18000000000000004e-51 < x < 4.3e14 or 5.5000000000000002e24 < x < 3.7999999999999998e39 or 7.0000000000000002e54 < x < 1.99999999999999997e77 or 1.55e111 < x < 1.30000000000000011e139

   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 y around inf 74.5%

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

   if 4.3e14 < x < 5.5000000000000002e24 or 3.7999999999999998e39 < x < 7.0000000000000002e54 or 1.99999999999999997e77 < x < 1.55e111 or 1.30000000000000011e139 < x

   1. Initial program 99.4%

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

   2. Taylor expanded in y around inf 99.4%

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

   3. Taylor expanded in y around 0 69.4%

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

      1. *-commutative 69.4%

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

   5. Simplified 69.4%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-163}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-51}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+14} \lor \neg \left(x \leq 5.5 \cdot 10^{+24}\right) \land \left(x \leq 3.8 \cdot 10^{+39} \lor \neg \left(x \leq 7 \cdot 10^{+54}\right) \land \left(x \leq 2 \cdot 10^{+77} \lor \neg \left(x \leq 1.55 \cdot 10^{+111}\right) \land x \leq 1.3 \cdot 10^{+139}\right)\right):\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 3: 62.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x}}\\ t_1 := \sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{if}\;x \leq 1.85 \cdot 10^{-166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+26} \lor \neg \left(x \leq 6 \cdot 10^{+39}\right) \land \left(x \leq 1.35 \cdot 10^{+55} \lor \neg \left(x \leq 3.5 \cdot 10^{+76}\right) \land \left(x \leq 1.75 \cdot 10^{+111} \lor \neg \left(x \leq 4.7 \cdot 10^{+138}\right)\right)\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (/ 0.1111111111111111 x))) (t_1 (* (sqrt x) (* y 3.0))))
   (if (<= x 1.85e-166)
     t_0
     (if (<= x 3.6e-163)
       t_1
       (if (<= x 8.5e-52)
         t_0
         (if (<= x 2.6e+14)
           t_1
           (if (or (<= x 5e+26)
                   (and (not (<= x 6e+39))
                        (or (<= x 1.35e+55)
                            (and (not (<= x 3.5e+76))
                                 (or (<= x 1.75e+111)
                                     (not (<= x 4.7e+138)))))))
             (* (sqrt x) -3.0)
             (* 3.0 (* y (sqrt x))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt((0.1111111111111111 / x));
	double t_1 = sqrt(x) * (y * 3.0);
	double tmp;
	if (x <= 1.85e-166) {
		tmp = t_0;
	} else if (x <= 3.6e-163) {
		tmp = t_1;
	} else if (x <= 8.5e-52) {
		tmp = t_0;
	} else if (x <= 2.6e+14) {
		tmp = t_1;
	} else if ((x <= 5e+26) || (!(x <= 6e+39) && ((x <= 1.35e+55) || (!(x <= 3.5e+76) && ((x <= 1.75e+111) || !(x <= 4.7e+138)))))) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = 3.0 * (y * sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((0.1111111111111111d0 / x))
    t_1 = sqrt(x) * (y * 3.0d0)
    if (x <= 1.85d-166) then
        tmp = t_0
    else if (x <= 3.6d-163) then
        tmp = t_1
    else if (x <= 8.5d-52) then
        tmp = t_0
    else if (x <= 2.6d+14) then
        tmp = t_1
    else if ((x <= 5d+26) .or. (.not. (x <= 6d+39)) .and. (x <= 1.35d+55) .or. (.not. (x <= 3.5d+76)) .and. (x <= 1.75d+111) .or. (.not. (x <= 4.7d+138))) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = 3.0d0 * (y * sqrt(x))
    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) * (y * 3.0);
	double tmp;
	if (x <= 1.85e-166) {
		tmp = t_0;
	} else if (x <= 3.6e-163) {
		tmp = t_1;
	} else if (x <= 8.5e-52) {
		tmp = t_0;
	} else if (x <= 2.6e+14) {
		tmp = t_1;
	} else if ((x <= 5e+26) || (!(x <= 6e+39) && ((x <= 1.35e+55) || (!(x <= 3.5e+76) && ((x <= 1.75e+111) || !(x <= 4.7e+138)))))) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = 3.0 * (y * Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((0.1111111111111111 / x))
	t_1 = math.sqrt(x) * (y * 3.0)
	tmp = 0
	if x <= 1.85e-166:
		tmp = t_0
	elif x <= 3.6e-163:
		tmp = t_1
	elif x <= 8.5e-52:
		tmp = t_0
	elif x <= 2.6e+14:
		tmp = t_1
	elif (x <= 5e+26) or (not (x <= 6e+39) and ((x <= 1.35e+55) or (not (x <= 3.5e+76) and ((x <= 1.75e+111) or not (x <= 4.7e+138))))):
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = 3.0 * (y * math.sqrt(x))
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(0.1111111111111111 / x))
	t_1 = Float64(sqrt(x) * Float64(y * 3.0))
	tmp = 0.0
	if (x <= 1.85e-166)
		tmp = t_0;
	elseif (x <= 3.6e-163)
		tmp = t_1;
	elseif (x <= 8.5e-52)
		tmp = t_0;
	elseif (x <= 2.6e+14)
		tmp = t_1;
	elseif ((x <= 5e+26) || (!(x <= 6e+39) && ((x <= 1.35e+55) || (!(x <= 3.5e+76) && ((x <= 1.75e+111) || !(x <= 4.7e+138))))))
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((0.1111111111111111 / x));
	t_1 = sqrt(x) * (y * 3.0);
	tmp = 0.0;
	if (x <= 1.85e-166)
		tmp = t_0;
	elseif (x <= 3.6e-163)
		tmp = t_1;
	elseif (x <= 8.5e-52)
		tmp = t_0;
	elseif (x <= 2.6e+14)
		tmp = t_1;
	elseif ((x <= 5e+26) || (~((x <= 6e+39)) && ((x <= 1.35e+55) || (~((x <= 3.5e+76)) && ((x <= 1.75e+111) || ~((x <= 4.7e+138)))))))
		tmp = sqrt(x) * -3.0;
	else
		tmp = 3.0 * (y * sqrt(x));
	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[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.85e-166], t$95$0, If[LessEqual[x, 3.6e-163], t$95$1, If[LessEqual[x, 8.5e-52], t$95$0, If[LessEqual[x, 2.6e+14], t$95$1, If[Or[LessEqual[x, 5e+26], And[N[Not[LessEqual[x, 6e+39]], $MachinePrecision], Or[LessEqual[x, 1.35e+55], And[N[Not[LessEqual[x, 3.5e+76]], $MachinePrecision], Or[LessEqual[x, 1.75e+111], N[Not[LessEqual[x, 4.7e+138]], $MachinePrecision]]]]]], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.8500000000000001e-166 or 3.5999999999999998e-163 < x < 8.50000000000000006e-52

   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* 99.5%

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

      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* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Applied egg-rr 28.4%

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

      1. associate-*r* 28.4%

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

      2. *-commutative 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in x around 0 83.7%

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

   if 1.8500000000000001e-166 < x < 3.5999999999999998e-163 or 8.50000000000000006e-52 < x < 2.6e14

   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* 99.1%

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

      2. associate--l+ 99.1%

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

      3. sub-neg 99.1%

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

      4. *-commutative 99.1%

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

      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 y around inf 65.3%

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

      1. *-commutative 65.3%

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

      2. associate-*l* 65.5%

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

      3. *-commutative 65.5%

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

   6. Simplified 65.5%

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

   if 2.6e14 < x < 5.0000000000000001e26 or 5.9999999999999999e39 < x < 1.34999999999999988e55 or 3.5e76 < x < 1.7500000000000001e111 or 4.6999999999999998e138 < x

   1. Initial program 99.4%

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

   2. Taylor expanded in y around inf 99.4%

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

   3. Taylor expanded in y around 0 69.4%

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

      1. *-commutative 69.4%

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

   5. Simplified 69.4%

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

   if 5.0000000000000001e26 < x < 5.9999999999999999e39 or 1.34999999999999988e55 < x < 3.5e76 or 1.7500000000000001e111 < x < 4.6999999999999998e138

   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. associate-*l* 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 82.8%

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-166}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-163}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+14}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;x \leq 5 \cdot 10^{+26} \lor \neg \left(x \leq 6 \cdot 10^{+39}\right) \land \left(x \leq 1.35 \cdot 10^{+55} \lor \neg \left(x \leq 3.5 \cdot 10^{+76}\right) \land \left(x \leq 1.75 \cdot 10^{+111} \lor \neg \left(x \leq 4.7 \cdot 10^{+138}\right)\right)\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \]

Alternative 4: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5)
   (* 3.0 (* y (sqrt x)))
   (if (<= y 1.05e+44)
     (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
     (* (sqrt x) (* y 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5) {
		tmp = 3.0 * (y * sqrt(x));
	} else if (y <= 1.05e+44) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = sqrt(x) * (y * 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 <= (-6.5d0)) then
        tmp = 3.0d0 * (y * sqrt(x))
    else if (y <= 1.05d+44) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = sqrt(x) * (y * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.5) {
		tmp = 3.0 * (y * Math.sqrt(x));
	} else if (y <= 1.05e+44) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = Math.sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.5:
		tmp = 3.0 * (y * math.sqrt(x))
	elif y <= 1.05e+44:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = math.sqrt(x) * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.5)
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	elseif (y <= 1.05e+44)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.5)
		tmp = 3.0 * (y * sqrt(x));
	elseif (y <= 1.05e+44)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = sqrt(x) * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.5], N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e+44], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.5

   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. associate-*l* 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. *-commutative 99.6%

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

      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 y around inf 76.4%

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

   if -6.5 < y < 1.04999999999999993e44

   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* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. sub-neg 97.7%

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

      3. associate-*r/ 97.9%

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

      4. metadata-eval 97.9%

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

      5. metadata-eval 97.9%

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

      6. *-commutative 97.9%

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

      7. *-commutative 97.9%

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

      8. associate-*l* 97.9%

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

      9. *-commutative 97.9%

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

      10. distribute-rgt-in 97.9%

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

      11. metadata-eval 97.9%

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

      12. associate-*l/ 97.9%

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

      13. metadata-eval 97.9%

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

   6. Simplified 97.9%

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

   if 1.04999999999999993e44 < y

   1. Initial program 99.5%

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

      1. 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 y around inf 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-*l* 76.6%

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

      3. *-commutative 76.6%

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

   6. Simplified 76.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 5: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.4:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.4)
   (* (sqrt x) (- (* y 3.0) 3.0))
   (if (<= y 1.5e+44)
     (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
     (* (sqrt x) (* y 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.4) {
		tmp = sqrt(x) * ((y * 3.0) - 3.0);
	} else if (y <= 1.5e+44) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = sqrt(x) * (y * 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 <= (-0.4d0)) then
        tmp = sqrt(x) * ((y * 3.0d0) - 3.0d0)
    else if (y <= 1.5d+44) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = sqrt(x) * (y * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -0.4) {
		tmp = Math.sqrt(x) * ((y * 3.0) - 3.0);
	} else if (y <= 1.5e+44) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = Math.sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.4:
		tmp = math.sqrt(x) * ((y * 3.0) - 3.0)
	elif y <= 1.5e+44:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = math.sqrt(x) * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.4)
		tmp = Float64(sqrt(x) * Float64(Float64(y * 3.0) - 3.0));
	elseif (y <= 1.5e+44)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.4)
		tmp = sqrt(x) * ((y * 3.0) - 3.0);
	elseif (y <= 1.5e+44)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = sqrt(x) * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.4], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(y * 3.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.5e+44], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.40000000000000002

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-lft-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/r* 99.4%

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

      10. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 78.3%

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

   if -0.40000000000000002 < y < 1.49999999999999993e44

   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* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. sub-neg 97.7%

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

      3. associate-*r/ 97.9%

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

      4. metadata-eval 97.9%

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

      5. metadata-eval 97.9%

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

      6. *-commutative 97.9%

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

      7. *-commutative 97.9%

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

      8. associate-*l* 97.9%

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

      9. *-commutative 97.9%

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

      10. distribute-rgt-in 97.9%

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

      11. metadata-eval 97.9%

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

      12. associate-*l/ 97.9%

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

      13. metadata-eval 97.9%

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

   6. Simplified 97.9%

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

   if 1.49999999999999993e44 < y

   1. Initial program 99.5%

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

      1. 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 y around inf 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-*l* 76.6%

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

      3. *-commutative 76.6%

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

   6. Simplified 76.6%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.4:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+44}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 6: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.7:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.7)
   (* (sqrt (* x 9.0)) (+ y -1.0))
   (if (<= y 5e+45)
     (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
     (* (sqrt x) (* y 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.7) {
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	} else if (y <= 5e+45) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = sqrt(x) * (y * 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 <= (-0.7d0)) then
        tmp = sqrt((x * 9.0d0)) * (y + (-1.0d0))
    else if (y <= 5d+45) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = sqrt(x) * (y * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -0.7) {
		tmp = Math.sqrt((x * 9.0)) * (y + -1.0);
	} else if (y <= 5e+45) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = Math.sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.7:
		tmp = math.sqrt((x * 9.0)) * (y + -1.0)
	elif y <= 5e+45:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = math.sqrt(x) * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.7)
		tmp = Float64(sqrt(Float64(x * 9.0)) * Float64(y + -1.0));
	elseif (y <= 5e+45)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.7)
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	elseif (y <= 5e+45)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = sqrt(x) * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.7], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+45], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.69999999999999996

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf 78.3%

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

      1. expm1-log1p-u 76.1%

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

      2. expm1-udef 47.4%

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

      3. *-commutative 47.4%

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

      4. metadata-eval 47.4%

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

      5. sqrt-prod 47.4%

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

   4. Applied egg-rr 47.4%

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

      1. expm1-def 76.1%

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

      2. expm1-log1p 78.4%

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

   6. Simplified 78.4%

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

   if -0.69999999999999996 < y < 5e45

   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* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 97.7%

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

      1. *-commutative 97.7%

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

      2. sub-neg 97.7%

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

      3. associate-*r/ 97.9%

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

      4. metadata-eval 97.9%

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

      5. metadata-eval 97.9%

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

      6. *-commutative 97.9%

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

      7. *-commutative 97.9%

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

      8. associate-*l* 97.9%

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

      9. *-commutative 97.9%

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

      10. distribute-rgt-in 97.9%

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

      11. metadata-eval 97.9%

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

      12. associate-*l/ 97.9%

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

      13. metadata-eval 97.9%

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

   6. Simplified 97.9%

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

   if 5e45 < y

   1. Initial program 99.5%

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

      1. 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 y around inf 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-*l* 76.6%

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

      3. *-commutative 76.6%

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

   6. Simplified 76.6%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.7:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \]

Alternative 7: 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 8: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. sub-neg 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-lft-in 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-/r* 99.4%

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

   10. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 9: 61.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3400:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3400.0) (sqrt (/ 0.1111111111111111 x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3400.0) {
		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 <= 3400.0d0) 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 <= 3400.0) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3400.0:
		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 <= 3400.0)
		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 <= 3400.0)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3400.0], 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 < 3400

   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* 99.4%

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

      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* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Applied egg-rr 30.4%

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

      1. associate-*r* 30.4%

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

      2. *-commutative 30.4%

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

   7. Simplified 30.4%

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

   8. Taylor expanded in x around 0 72.8%

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

   if 3400 < x

   1. Initial program 99.4%

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

   2. Taylor expanded in y around inf 98.9%

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

   3. Taylor expanded in y around 0 54.7%

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

      1. *-commutative 54.7%

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

   5. Simplified 54.7%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3400:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 10: 3.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

2. Taylor expanded in y around inf 61.8%

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

3. Taylor expanded in y around 0 27.6%

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

   1. *-commutative 27.6%

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

5. Simplified 27.6%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 3.0%

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

   3. pow2 3.0%

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt{x} \cdot -3\right)}^{2}}} \]

7. Applied egg-rr 3.0%

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

   1. unpow2 3.0%

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

   2. swap-sqr 3.0%

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

   3. unpow1/2 3.0%

      \[\leadsto \sqrt{\left(\color{blue}{{x}^{0.5}} \cdot \sqrt{x}\right) \cdot \left(-3 \cdot -3\right)} \]

   4. metadata-eval 3.0%

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

   5. unpow1/2 3.0%

      \[\leadsto \sqrt{\left({x}^{\left(\frac{1}{2}\right)} \cdot \color{blue}{{x}^{0.5}}\right) \cdot \left(-3 \cdot -3\right)} \]

   6. metadata-eval 3.0%

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

   7. sqr-pow 3.0%

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

   8. unpow1 3.0%

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

   9. metadata-eval 3.0%

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

9. Simplified 3.0%

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

10. Final simplification 3.0%

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

Alternative 11: 38.3% 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%

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

5. Applied egg-rr 21.1%

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

   1. associate-*r* 21.1%

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

   2. *-commutative 21.1%

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

7. Simplified 21.1%

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

8. Taylor expanded in x around 0 38.5%

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

9. Final simplification 38.5%

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

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 2023322 
(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)))