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

Percentage Accurate: 99.4% → 99.4%

Time: 8.5s

Alternatives: 7

Speedup: N/A×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. 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. associate-/r* 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in y around 0 99.3%

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

   1. *-commutative 99.3%

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

   2. sub-neg 99.3%

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

   3. associate-*r/ 99.4%

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

   4. metadata-eval 99.4%

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

   5. metadata-eval 99.4%

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

   6. associate-*l* 99.4%

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

   7. distribute-rgt-in 99.4%

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

   8. *-commutative 99.4%

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

   9. *-commutative 99.4%

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

   10. associate-*l* 99.4%

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

   11. *-commutative 99.4%

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

   12. associate-+l+ 99.4%

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

   13. distribute-rgt-in 99.4%

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

   14. +-commutative 99.4%

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

   15. distribute-lft-in 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 10600000 \lor \neg \left(x \leq 1.02 \cdot 10^{+155}\right) \land x \leq 10^{+254}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.35e-27)
   (* (sqrt x) (/ 0.3333333333333333 x))
   (if (or (<= x 10600000.0) (and (not (<= x 1.02e+155)) (<= x 1e+254)))
     (* 3.0 (* (sqrt x) y))
     (* (sqrt x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.35e-27) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else if ((x <= 10600000.0) || (!(x <= 1.02e+155) && (x <= 1e+254))) {
		tmp = 3.0 * (sqrt(x) * y);
	} 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 <= 1.35d-27) then
        tmp = sqrt(x) * (0.3333333333333333d0 / x)
    else if ((x <= 10600000.0d0) .or. (.not. (x <= 1.02d+155)) .and. (x <= 1d+254)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    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 <= 1.35e-27) {
		tmp = Math.sqrt(x) * (0.3333333333333333 / x);
	} else if ((x <= 10600000.0) || (!(x <= 1.02e+155) && (x <= 1e+254))) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.35e-27:
		tmp = math.sqrt(x) * (0.3333333333333333 / x)
	elif (x <= 10600000.0) or (not (x <= 1.02e+155) and (x <= 1e+254)):
		tmp = 3.0 * (math.sqrt(x) * y)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.35e-27)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	elseif ((x <= 10600000.0) || (!(x <= 1.02e+155) && (x <= 1e+254)))
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.35e-27)
		tmp = sqrt(x) * (0.3333333333333333 / x);
	elseif ((x <= 10600000.0) || (~((x <= 1.02e+155)) && (x <= 1e+254)))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.35e-27], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 10600000.0], And[N[Not[LessEqual[x, 1.02e+155]], $MachinePrecision], LessEqual[x, 1e+254]]], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.34999999999999994e-27

   1. Initial program 99.2%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 83.5%

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

   if 1.34999999999999994e-27 < x < 1.06e7 or 1.02e155 < x < 9.9999999999999994e253

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 64.0%

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

   if 1.06e7 < x < 1.02e155 or 9.9999999999999994e253 < x

   1. Initial program 99.5%

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

   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 98.4%

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

   5. Taylor expanded in y around 0 61.2%

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

      1. *-commutative 61.2%

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

   7. Simplified 61.2%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 10600000 \lor \neg \left(x \leq 1.02 \cdot 10^{+155}\right) \land x \leq 10^{+254}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 3: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 12500000:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+154} \lor \neg \left(x \leq 3.6 \cdot 10^{+254}\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.05e-27)
   (* (sqrt x) (/ 0.3333333333333333 x))
   (if (<= x 12500000.0)
     (* y (* 3.0 (sqrt x)))
     (if (or (<= x 4.8e+154) (not (<= x 3.6e+254)))
       (* (sqrt x) -3.0)
       (* 3.0 (* (sqrt x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.05e-27) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else if (x <= 12500000.0) {
		tmp = y * (3.0 * sqrt(x));
	} else if ((x <= 4.8e+154) || !(x <= 3.6e+254)) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = 3.0 * (sqrt(x) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.05d-27) then
        tmp = sqrt(x) * (0.3333333333333333d0 / x)
    else if (x <= 12500000.0d0) then
        tmp = y * (3.0d0 * sqrt(x))
    else if ((x <= 4.8d+154) .or. (.not. (x <= 3.6d+254))) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = 3.0d0 * (sqrt(x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.05e-27) {
		tmp = Math.sqrt(x) * (0.3333333333333333 / x);
	} else if (x <= 12500000.0) {
		tmp = y * (3.0 * Math.sqrt(x));
	} else if ((x <= 4.8e+154) || !(x <= 3.6e+254)) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.05e-27:
		tmp = math.sqrt(x) * (0.3333333333333333 / x)
	elif x <= 12500000.0:
		tmp = y * (3.0 * math.sqrt(x))
	elif (x <= 4.8e+154) or not (x <= 3.6e+254):
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.05e-27)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	elseif (x <= 12500000.0)
		tmp = Float64(y * Float64(3.0 * sqrt(x)));
	elseif ((x <= 4.8e+154) || !(x <= 3.6e+254))
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.05e-27)
		tmp = sqrt(x) * (0.3333333333333333 / x);
	elseif (x <= 12500000.0)
		tmp = y * (3.0 * sqrt(x));
	elseif ((x <= 4.8e+154) || ~((x <= 3.6e+254)))
		tmp = sqrt(x) * -3.0;
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.05e-27], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 12500000.0], N[(y * N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 4.8e+154], N[Not[LessEqual[x, 3.6e+254]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.05000000000000008e-27

   1. Initial program 99.2%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 83.5%

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

   if 1.05000000000000008e-27 < x < 1.25e7

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-/r* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 57.8%

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

   if 1.25e7 < x < 4.8000000000000003e154 or 3.59999999999999977e254 < x

   1. Initial program 99.5%

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

   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 98.4%

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

   5. Taylor expanded in y around 0 61.2%

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

      1. *-commutative 61.2%

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

   7. Simplified 61.2%

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

   if 4.8000000000000003e154 < x < 3.59999999999999977e254

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 67.9%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.05 \cdot 10^{-27}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 12500000:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+154} \lor \neg \left(x \leq 3.6 \cdot 10^{+254}\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 4: 60.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{elif}\;x \leq 7000000:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+153} \lor \neg \left(x \leq 1.08 \cdot 10^{+254}\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9e-29)
   (/ (sqrt x) (* 3.0 x))
   (if (<= x 7000000.0)
     (* y (* 3.0 (sqrt x)))
     (if (or (<= x 1.35e+153) (not (<= x 1.08e+254)))
       (* (sqrt x) -3.0)
       (* 3.0 (* (sqrt x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9e-29) {
		tmp = sqrt(x) / (3.0 * x);
	} else if (x <= 7000000.0) {
		tmp = y * (3.0 * sqrt(x));
	} else if ((x <= 1.35e+153) || !(x <= 1.08e+254)) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = 3.0 * (sqrt(x) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 9d-29) then
        tmp = sqrt(x) / (3.0d0 * x)
    else if (x <= 7000000.0d0) then
        tmp = y * (3.0d0 * sqrt(x))
    else if ((x <= 1.35d+153) .or. (.not. (x <= 1.08d+254))) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = 3.0d0 * (sqrt(x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 9e-29) {
		tmp = Math.sqrt(x) / (3.0 * x);
	} else if (x <= 7000000.0) {
		tmp = y * (3.0 * Math.sqrt(x));
	} else if ((x <= 1.35e+153) || !(x <= 1.08e+254)) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 9e-29:
		tmp = math.sqrt(x) / (3.0 * x)
	elif x <= 7000000.0:
		tmp = y * (3.0 * math.sqrt(x))
	elif (x <= 1.35e+153) or not (x <= 1.08e+254):
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9e-29)
		tmp = Float64(sqrt(x) / Float64(3.0 * x));
	elseif (x <= 7000000.0)
		tmp = Float64(y * Float64(3.0 * sqrt(x)));
	elseif ((x <= 1.35e+153) || !(x <= 1.08e+254))
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 9e-29)
		tmp = sqrt(x) / (3.0 * x);
	elseif (x <= 7000000.0)
		tmp = y * (3.0 * sqrt(x));
	elseif ((x <= 1.35e+153) || ~((x <= 1.08e+254)))
		tmp = sqrt(x) * -3.0;
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9e-29], N[(N[Sqrt[x], $MachinePrecision] / N[(3.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7000000.0], N[(y * N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 1.35e+153], N[Not[LessEqual[x, 1.08e+254]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.9999999999999996e-29

   1. Initial program 99.2%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 83.5%

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

      1. expm1-log1p-u 77.7%

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

      2. expm1-udef 77.7%

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

   6. Applied egg-rr 77.7%

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

      1. expm1-def 77.7%

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

      2. expm1-log1p 83.5%

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

      3. associate-*r/ 83.5%

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

      4. associate-/l* 83.6%

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

      5. metadata-eval 83.6%

         \[\leadsto \frac{\sqrt{x}}{\frac{x}{\color{blue}{\frac{1}{3}}}} \]

      6. associate-/l* 83.7%

         \[\leadsto \frac{\sqrt{x}}{\color{blue}{\frac{x \cdot 3}{1}}} \]

      7. /-rgt-identity 83.7%

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

   8. Simplified 83.7%

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

   if 8.9999999999999996e-29 < x < 7e6

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-/r* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 57.8%

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

   if 7e6 < x < 1.35e153 or 1.08000000000000001e254 < x

   1. Initial program 99.5%

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

   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 98.4%

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

   5. Taylor expanded in y around 0 61.2%

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

      1. *-commutative 61.2%

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

   7. Simplified 61.2%

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

   if 1.35e153 < x < 1.08000000000000001e254

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 67.9%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{elif}\;x \leq 7000000:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+153} \lor \neg \left(x \leq 1.08 \cdot 10^{+254}\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 5: 61.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.0) (not (<= y 1.0)))
   (* 3.0 (* (sqrt x) y))
   (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = 3.0 * (sqrt(x) * y);
	} 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 ((y <= (-1.0d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.0) || !(y <= 1.0)) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.0) or not (y <= 1.0):
		tmp = 3.0 * (math.sqrt(x) * y)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.0) || !(y <= 1.0))
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.0) || ~((y <= 1.0)))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1 or 1 < y

   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%

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

      2. associate-/r* 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 69.3%

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

   if -1 < y < 1

   1. Initial program 99.4%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around inf 45.2%

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

   5. Taylor expanded in y around 0 44.3%

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

      1. *-commutative 44.3%

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

   7. Simplified 44.3%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 6: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.25e-29) (/ (sqrt x) (* 3.0 x)) (* 3.0 (* (sqrt x) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.25e-29) {
		tmp = sqrt(x) / (3.0 * x);
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.25d-29) then
        tmp = sqrt(x) / (3.0d0 * x)
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.25e-29) {
		tmp = Math.sqrt(x) / (3.0 * x);
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.25e-29:
		tmp = math.sqrt(x) / (3.0 * x)
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.25e-29)
		tmp = Float64(sqrt(x) / Float64(3.0 * x));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.25e-29)
		tmp = sqrt(x) / (3.0 * x);
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.25e-29], N[(N[Sqrt[x], $MachinePrecision] / N[(3.0 * x), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.24999999999999996e-29

   1. Initial program 99.2%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 83.5%

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

      1. expm1-log1p-u 77.7%

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

      2. expm1-udef 77.7%

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

   6. Applied egg-rr 77.7%

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

      1. expm1-def 77.7%

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

      2. expm1-log1p 83.5%

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

      3. associate-*r/ 83.5%

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

      4. associate-/l* 83.6%

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

      5. metadata-eval 83.6%

         \[\leadsto \frac{\sqrt{x}}{\frac{x}{\color{blue}{\frac{1}{3}}}} \]

      6. associate-/l* 83.7%

         \[\leadsto \frac{\sqrt{x}}{\color{blue}{\frac{x \cdot 3}{1}}} \]

      7. /-rgt-identity 83.7%

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

   8. Simplified 83.7%

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

   if 1.24999999999999996e-29 < 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. associate--l+ 99.5%

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

      2. associate-/r* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 99.5%

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

      1. *-commutative 99.5%

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

      2. sub-neg 99.5%

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

      3. associate-*r/ 99.5%

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

      4. metadata-eval 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*l* 99.6%

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

      7. distribute-rgt-in 99.6%

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

      8. *-commutative 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-*l* 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-+l+ 99.5%

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

      13. distribute-rgt-in 99.5%

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

      14. +-commutative 99.5%

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

      15. distribute-lft-in 99.5%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 93.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 7: 26.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot -3 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

4. Taylor expanded in x around inf 56.6%

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

5. Taylor expanded in y around 0 24.3%

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

   1. *-commutative 24.3%

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

7. Simplified 24.3%

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

8. Final simplification 24.3%

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

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 2023192 
(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)))