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

Percentage Accurate: 99.4% → 99.4%

Time: 12.2s

Alternatives: 12

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

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

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 99.5%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
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. sub-neg 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. metadata-eval 99.4%

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

   3. sqrt-prod 99.6%

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

   4. pow1/2 99.6%

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

5. Applied egg-rr 99.6%

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

   1. unpow1/2 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 61.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 1800000000000:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+58} \lor \neg \left(x \leq 5.2 \cdot 10^{+77}\right) \land \left(x \leq 3.8 \cdot 10^{+114} \lor \neg \left(x \leq 1.6 \cdot 10^{+169}\right) \land \left(x \leq 3.2 \cdot 10^{+244} \lor \neg \left(x \leq 4.5 \cdot 10^{+273}\right)\right)\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8e-66)
   (* (sqrt x) (/ 0.3333333333333333 x))
   (if (<= x 1800000000000.0)
     (* y (* 3.0 (sqrt x)))
     (if (or (<= x 1.25e+58)
             (and (not (<= x 5.2e+77))
                  (or (<= x 3.8e+114)
                      (and (not (<= x 1.6e+169))
                           (or (<= x 3.2e+244) (not (<= x 4.5e+273)))))))
       (* (sqrt x) -3.0)
       (* (sqrt (* x 9.0)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8e-66) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else if (x <= 1800000000000.0) {
		tmp = y * (3.0 * sqrt(x));
	} else if ((x <= 1.25e+58) || (!(x <= 5.2e+77) && ((x <= 3.8e+114) || (!(x <= 1.6e+169) && ((x <= 3.2e+244) || !(x <= 4.5e+273)))))) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = sqrt((x * 9.0)) * 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 <= 8d-66) then
        tmp = sqrt(x) * (0.3333333333333333d0 / x)
    else if (x <= 1800000000000.0d0) then
        tmp = y * (3.0d0 * sqrt(x))
    else if ((x <= 1.25d+58) .or. (.not. (x <= 5.2d+77)) .and. (x <= 3.8d+114) .or. (.not. (x <= 1.6d+169)) .and. (x <= 3.2d+244) .or. (.not. (x <= 4.5d+273))) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = sqrt((x * 9.0d0)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 8e-66) {
		tmp = Math.sqrt(x) * (0.3333333333333333 / x);
	} else if (x <= 1800000000000.0) {
		tmp = y * (3.0 * Math.sqrt(x));
	} else if ((x <= 1.25e+58) || (!(x <= 5.2e+77) && ((x <= 3.8e+114) || (!(x <= 1.6e+169) && ((x <= 3.2e+244) || !(x <= 4.5e+273)))))) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = Math.sqrt((x * 9.0)) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8e-66:
		tmp = math.sqrt(x) * (0.3333333333333333 / x)
	elif x <= 1800000000000.0:
		tmp = y * (3.0 * math.sqrt(x))
	elif (x <= 1.25e+58) or (not (x <= 5.2e+77) and ((x <= 3.8e+114) or (not (x <= 1.6e+169) and ((x <= 3.2e+244) or not (x <= 4.5e+273))))):
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = math.sqrt((x * 9.0)) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8e-66)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	elseif (x <= 1800000000000.0)
		tmp = Float64(y * Float64(3.0 * sqrt(x)));
	elseif ((x <= 1.25e+58) || (!(x <= 5.2e+77) && ((x <= 3.8e+114) || (!(x <= 1.6e+169) && ((x <= 3.2e+244) || !(x <= 4.5e+273))))))
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8e-66)
		tmp = sqrt(x) * (0.3333333333333333 / x);
	elseif (x <= 1800000000000.0)
		tmp = y * (3.0 * sqrt(x));
	elseif ((x <= 1.25e+58) || (~((x <= 5.2e+77)) && ((x <= 3.8e+114) || (~((x <= 1.6e+169)) && ((x <= 3.2e+244) || ~((x <= 4.5e+273)))))))
		tmp = sqrt(x) * -3.0;
	else
		tmp = sqrt((x * 9.0)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8e-66], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1800000000000.0], N[(y * N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 1.25e+58], And[N[Not[LessEqual[x, 5.2e+77]], $MachinePrecision], Or[LessEqual[x, 3.8e+114], And[N[Not[LessEqual[x, 1.6e+169]], $MachinePrecision], Or[LessEqual[x, 3.2e+244], N[Not[LessEqual[x, 4.5e+273]], $MachinePrecision]]]]]], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.9999999999999998e-66

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 98.3%

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

      3. +-commutative 98.3%

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

      4. associate--l+ 98.3%

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

      5. +-commutative 98.3%

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

      6. distribute-rgt-in 98.4%

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

      7. *-commutative 98.4%

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

      8. fma-def 98.4%

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

      9. sub-neg 98.4%

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

      10. metadata-eval 98.4%

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

      11. associate-*l/ 98.5%

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

      12. metadata-eval 98.5%

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

      13. *-commutative 98.5%

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

      14. associate-/r* 98.3%

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

      15. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in x around 0 88.3%

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

   if 7.9999999999999998e-66 < x < 1.8e12

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 62.6%

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

      1. expm1-log1p-u 23.3%

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

      2. expm1-udef 23.2%

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

      3. associate-*r* 23.2%

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

      4. *-commutative 23.2%

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

      5. metadata-eval 23.2%

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

      6. sqrt-prod 23.2%

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

   4. Applied egg-rr 23.2%

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

      1. expm1-def 23.3%

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

      2. expm1-log1p 62.8%

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

      3. *-commutative 62.8%

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

   6. Simplified 62.8%

      \[\leadsto \color{blue}{y \cdot \sqrt{x \cdot 9}} \]
   7. Step-by-step derivation

      1. sqrt-prod 62.8%

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

      2. metadata-eval 62.8%

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

   8. Applied egg-rr 62.8%

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

   if 1.8e12 < x < 1.24999999999999996e58 or 5.2000000000000004e77 < x < 3.8000000000000001e114 or 1.5999999999999999e169 < x < 3.2000000000000002e244 or 4.49999999999999993e273 < x

   1. Initial program 99.6%

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

      1. 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. sub-neg 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.8%

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

      4. pow1/2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow1/2 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf 99.7%

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

   9. Taylor expanded in y around 0 71.6%

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

       1. *-commutative 71.6%

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

   11. Simplified 71.6%

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

   if 1.24999999999999996e58 < x < 5.2000000000000004e77 or 3.8000000000000001e114 < x < 1.5999999999999999e169 or 3.2000000000000002e244 < x < 4.49999999999999993e273

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 74.5%

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

      1. expm1-log1p-u 33.6%

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

      2. expm1-udef 33.4%

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

      3. associate-*r* 33.4%

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

      4. *-commutative 33.4%

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

      5. metadata-eval 33.4%

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

      6. sqrt-prod 33.4%

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

   4. Applied egg-rr 33.4%

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

      1. expm1-def 33.6%

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

      2. expm1-log1p 74.6%

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

      3. *-commutative 74.6%

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

   6. Simplified 74.6%

      \[\leadsto \color{blue}{y \cdot \sqrt{x \cdot 9}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-66}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 1800000000000:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+58} \lor \neg \left(x \leq 5.2 \cdot 10^{+77}\right) \land \left(x \leq 3.8 \cdot 10^{+114} \lor \neg \left(x \leq 1.6 \cdot 10^{+169}\right) \land \left(x \leq 3.2 \cdot 10^{+244} \lor \neg \left(x \leq 4.5 \cdot 10^{+273}\right)\right)\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \]

Alternative 3: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 350000000000:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+59} \lor \neg \left(x \leq 3.6 \cdot 10^{+77}\right) \land \left(x \leq 5 \cdot 10^{+114} \lor \neg \left(x \leq 5.2 \cdot 10^{+168}\right) \land \left(x \leq 3 \cdot 10^{+244} \lor \neg \left(x \leq 4.8 \cdot 10^{+273}\right)\right)\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8e-66)
   (/ (sqrt x) (* x 3.0))
   (if (<= x 350000000000.0)
     (* y (* 3.0 (sqrt x)))
     (if (or (<= x 2.8e+59)
             (and (not (<= x 3.6e+77))
                  (or (<= x 5e+114)
                      (and (not (<= x 5.2e+168))
                           (or (<= x 3e+244) (not (<= x 4.8e+273)))))))
       (* (sqrt x) -3.0)
       (* (sqrt (* x 9.0)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8e-66) {
		tmp = sqrt(x) / (x * 3.0);
	} else if (x <= 350000000000.0) {
		tmp = y * (3.0 * sqrt(x));
	} else if ((x <= 2.8e+59) || (!(x <= 3.6e+77) && ((x <= 5e+114) || (!(x <= 5.2e+168) && ((x <= 3e+244) || !(x <= 4.8e+273)))))) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = sqrt((x * 9.0)) * 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 <= 8d-66) then
        tmp = sqrt(x) / (x * 3.0d0)
    else if (x <= 350000000000.0d0) then
        tmp = y * (3.0d0 * sqrt(x))
    else if ((x <= 2.8d+59) .or. (.not. (x <= 3.6d+77)) .and. (x <= 5d+114) .or. (.not. (x <= 5.2d+168)) .and. (x <= 3d+244) .or. (.not. (x <= 4.8d+273))) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = sqrt((x * 9.0d0)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 8e-66) {
		tmp = Math.sqrt(x) / (x * 3.0);
	} else if (x <= 350000000000.0) {
		tmp = y * (3.0 * Math.sqrt(x));
	} else if ((x <= 2.8e+59) || (!(x <= 3.6e+77) && ((x <= 5e+114) || (!(x <= 5.2e+168) && ((x <= 3e+244) || !(x <= 4.8e+273)))))) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = Math.sqrt((x * 9.0)) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8e-66:
		tmp = math.sqrt(x) / (x * 3.0)
	elif x <= 350000000000.0:
		tmp = y * (3.0 * math.sqrt(x))
	elif (x <= 2.8e+59) or (not (x <= 3.6e+77) and ((x <= 5e+114) or (not (x <= 5.2e+168) and ((x <= 3e+244) or not (x <= 4.8e+273))))):
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = math.sqrt((x * 9.0)) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8e-66)
		tmp = Float64(sqrt(x) / Float64(x * 3.0));
	elseif (x <= 350000000000.0)
		tmp = Float64(y * Float64(3.0 * sqrt(x)));
	elseif ((x <= 2.8e+59) || (!(x <= 3.6e+77) && ((x <= 5e+114) || (!(x <= 5.2e+168) && ((x <= 3e+244) || !(x <= 4.8e+273))))))
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8e-66)
		tmp = sqrt(x) / (x * 3.0);
	elseif (x <= 350000000000.0)
		tmp = y * (3.0 * sqrt(x));
	elseif ((x <= 2.8e+59) || (~((x <= 3.6e+77)) && ((x <= 5e+114) || (~((x <= 5.2e+168)) && ((x <= 3e+244) || ~((x <= 4.8e+273)))))))
		tmp = sqrt(x) * -3.0;
	else
		tmp = sqrt((x * 9.0)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8e-66], N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 350000000000.0], N[(y * N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 2.8e+59], And[N[Not[LessEqual[x, 3.6e+77]], $MachinePrecision], Or[LessEqual[x, 5e+114], And[N[Not[LessEqual[x, 5.2e+168]], $MachinePrecision], Or[LessEqual[x, 3e+244], N[Not[LessEqual[x, 4.8e+273]], $MachinePrecision]]]]]], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.9999999999999998e-66

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 98.3%

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

      3. +-commutative 98.3%

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

      4. associate--l+ 98.3%

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

      5. +-commutative 98.3%

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

      6. distribute-rgt-in 98.4%

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

      7. *-commutative 98.4%

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

      8. fma-def 98.4%

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

      9. sub-neg 98.4%

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

      10. metadata-eval 98.4%

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

      11. associate-*l/ 98.5%

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

      12. metadata-eval 98.5%

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

      13. *-commutative 98.5%

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

      14. associate-/r* 98.3%

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

      15. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in x around 0 88.3%

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

      1. expm1-log1p-u 82.1%

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

      2. expm1-udef 82.1%

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

      3. clear-num 82.1%

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

      4. un-div-inv 82.1%

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

      5. div-inv 82.1%

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

      6. metadata-eval 82.1%

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

   6. Applied egg-rr 82.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x}}{x \cdot 3}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 82.1%

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

      2. expm1-log1p 88.5%

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

   8. Simplified 88.5%

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

   if 7.9999999999999998e-66 < x < 3.5e11

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 62.6%

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

      1. expm1-log1p-u 23.3%

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

      2. expm1-udef 23.2%

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

      3. associate-*r* 23.2%

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

      4. *-commutative 23.2%

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

      5. metadata-eval 23.2%

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

      6. sqrt-prod 23.2%

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

   4. Applied egg-rr 23.2%

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

      1. expm1-def 23.3%

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

      2. expm1-log1p 62.8%

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

      3. *-commutative 62.8%

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

   6. Simplified 62.8%

      \[\leadsto \color{blue}{y \cdot \sqrt{x \cdot 9}} \]
   7. Step-by-step derivation

      1. sqrt-prod 62.8%

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

      2. metadata-eval 62.8%

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

   8. Applied egg-rr 62.8%

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

   if 3.5e11 < x < 2.7999999999999998e59 or 3.5999999999999998e77 < x < 5.0000000000000001e114 or 5.2e168 < x < 2.9999999999999998e244 or 4.8000000000000003e273 < x

   1. Initial program 99.6%

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

      1. 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. sub-neg 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.8%

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

      4. pow1/2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow1/2 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf 99.7%

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

   9. Taylor expanded in y around 0 71.6%

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

       1. *-commutative 71.6%

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

   11. Simplified 71.6%

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

   if 2.7999999999999998e59 < x < 3.5999999999999998e77 or 5.0000000000000001e114 < x < 5.2e168 or 2.9999999999999998e244 < x < 4.8000000000000003e273

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 74.5%

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

      1. expm1-log1p-u 33.6%

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

      2. expm1-udef 33.4%

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

      3. associate-*r* 33.4%

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

      4. *-commutative 33.4%

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

      5. metadata-eval 33.4%

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

      6. sqrt-prod 33.4%

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

   4. Applied egg-rr 33.4%

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

      1. expm1-def 33.6%

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

      2. expm1-log1p 74.6%

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

      3. *-commutative 74.6%

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

   6. Simplified 74.6%

      \[\leadsto \color{blue}{y \cdot \sqrt{x \cdot 9}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{elif}\;x \leq 350000000000:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+59} \lor \neg \left(x \leq 3.6 \cdot 10^{+77}\right) \land \left(x \leq 5 \cdot 10^{+114} \lor \neg \left(x \leq 5.2 \cdot 10^{+168}\right) \land \left(x \leq 3 \cdot 10^{+244} \lor \neg \left(x \leq 4.8 \cdot 10^{+273}\right)\right)\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112)
   (* (sqrt x) (+ (/ 0.3333333333333333 x) (* y 3.0)))
   (* (sqrt (* x 9.0)) (+ y -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + (y * 3.0));
	} else {
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.112d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (y * 3.0d0))
    else
        tmp = sqrt((x * 9.0d0)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + (y * 3.0));
	} else {
		tmp = Math.sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + (y * 3.0))
	else:
		tmp = math.sqrt((x * 9.0)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + Float64(y * 3.0)));
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + (y * 3.0));
	else
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   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* 98.6%

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

      3. +-commutative 98.6%

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

      4. associate--l+ 98.6%

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

      5. +-commutative 98.6%

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

      6. distribute-rgt-in 98.6%

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

      7. *-commutative 98.6%

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

      8. fma-def 98.6%

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

      9. sub-neg 98.6%

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

      10. metadata-eval 98.6%

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

      11. associate-*l/ 98.7%

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

      12. metadata-eval 98.7%

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

      13. *-commutative 98.7%

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

      14. associate-/r* 98.5%

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

      15. metadata-eval 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in y around 0 98.5%

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

      1. associate-*r/ 98.5%

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

      2. metadata-eval 98.5%

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

      3. associate-+r- 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in y around inf 97.7%

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

   if 0.112000000000000002 < x

   1. Initial program 99.6%

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

      1. 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. sub-neg 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.8%

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

      4. pow1/2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow1/2 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf 98.7%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 5: 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 99.5%

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

   1. associate-*l* 99.4%

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

   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 6: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0074 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.0074) (not (<= y 1.0)))
   (* 3.0 (* y (sqrt x)))
   (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.0074) || !(y <= 1.0)) {
		tmp = 3.0 * (y * sqrt(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 ((y <= (-0.0074d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = 3.0d0 * (y * sqrt(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 ((y <= -0.0074) || !(y <= 1.0)) {
		tmp = 3.0 * (y * Math.sqrt(x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.0074) or not (y <= 1.0):
		tmp = 3.0 * (y * math.sqrt(x))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.0074) || !(y <= 1.0))
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.0074) || ~((y <= 1.0)))
		tmp = 3.0 * (y * sqrt(x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.0074], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0074000000000000003 or 1 < 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. Taylor expanded in y around inf 71.5%

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

   if -0.0074000000000000003 < y < 1

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.5%

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

      4. pow1/2 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 49.4%

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

   9. Taylor expanded in y around 0 48.3%

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

       1. *-commutative 48.3%

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

   11. Simplified 48.3%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0074 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 7: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0074 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -0.0074) (not (<= y 1.0)))
   (* (sqrt (* x 9.0)) y)
   (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -0.0074) || !(y <= 1.0)) {
		tmp = sqrt((x * 9.0)) * 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 <= (-0.0074d0)) .or. (.not. (y <= 1.0d0))) then
        tmp = sqrt((x * 9.0d0)) * 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 <= -0.0074) || !(y <= 1.0)) {
		tmp = Math.sqrt((x * 9.0)) * y;
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -0.0074) or not (y <= 1.0):
		tmp = math.sqrt((x * 9.0)) * y
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -0.0074) || !(y <= 1.0))
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -0.0074) || ~((y <= 1.0)))
		tmp = sqrt((x * 9.0)) * y;
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -0.0074], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.0074000000000000003 or 1 < 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. Taylor expanded in y around inf 71.5%

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

      1. expm1-log1p-u 29.6%

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

      2. expm1-udef 29.5%

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

      3. associate-*r* 29.5%

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

      4. *-commutative 29.5%

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

      5. metadata-eval 29.5%

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

      6. sqrt-prod 29.5%

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

   4. Applied egg-rr 29.5%

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

      1. expm1-def 29.6%

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

      2. expm1-log1p 71.6%

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

      3. *-commutative 71.6%

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

   6. Simplified 71.6%

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

   if -0.0074000000000000003 < y < 1

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.5%

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

      4. pow1/2 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 49.4%

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

   9. Taylor expanded in y around 0 48.3%

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

       1. *-commutative 48.3%

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

   11. Simplified 48.3%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0074 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 8: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.0074:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.0074)
   (* y (* 3.0 (sqrt x)))
   (if (<= y 1.0) (* (sqrt x) -3.0) (* (sqrt (* x 9.0)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.0074) {
		tmp = y * (3.0 * sqrt(x));
	} else if (y <= 1.0) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = sqrt((x * 9.0)) * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-0.0074d0)) then
        tmp = y * (3.0d0 * sqrt(x))
    else if (y <= 1.0d0) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = sqrt((x * 9.0d0)) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -0.0074) {
		tmp = y * (3.0 * Math.sqrt(x));
	} else if (y <= 1.0) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = Math.sqrt((x * 9.0)) * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.0074:
		tmp = y * (3.0 * math.sqrt(x))
	elif y <= 1.0:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = math.sqrt((x * 9.0)) * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.0074)
		tmp = Float64(y * Float64(3.0 * sqrt(x)));
	elseif (y <= 1.0)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.0074)
		tmp = y * (3.0 * sqrt(x));
	elseif (y <= 1.0)
		tmp = sqrt(x) * -3.0;
	else
		tmp = sqrt((x * 9.0)) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.0074], N[(y * N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0074000000000000003

   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 74.8%

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

      1. expm1-log1p-u 0.1%

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

      2. expm1-udef 0.2%

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

      3. associate-*r* 0.2%

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

      4. *-commutative 0.2%

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

      5. metadata-eval 0.2%

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

      6. sqrt-prod 0.2%

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

   4. Applied egg-rr 0.2%

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

      1. expm1-def 0.1%

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

      2. expm1-log1p 75.0%

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

      3. *-commutative 75.0%

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

   6. Simplified 75.0%

      \[\leadsto \color{blue}{y \cdot \sqrt{x \cdot 9}} \]
   7. Step-by-step derivation

      1. sqrt-prod 75.0%

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

      2. metadata-eval 75.0%

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

   8. Applied egg-rr 75.0%

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

   if -0.0074000000000000003 < y < 1

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.5%

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

      4. pow1/2 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in x around inf 49.4%

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

   9. Taylor expanded in y around 0 48.3%

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

       1. *-commutative 48.3%

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

   11. Simplified 48.3%

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

   if 1 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in y around inf 67.7%

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

      1. expm1-log1p-u 62.9%

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

      2. expm1-udef 62.5%

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

      3. associate-*r* 62.5%

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

      4. *-commutative 62.5%

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

      5. metadata-eval 62.5%

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

      6. sqrt-prod 62.5%

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

   4. Applied egg-rr 62.5%

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

      1. expm1-def 62.9%

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

      2. expm1-log1p 67.8%

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

      3. *-commutative 67.8%

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

   6. Simplified 67.8%

      \[\leadsto \color{blue}{y \cdot \sqrt{x \cdot 9}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0074:\\ \;\;\;\;y \cdot \left(3 \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \]

Alternative 9: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6e-37) (/ (sqrt x) (* x 3.0)) (* (sqrt x) (+ (* y 3.0) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6e-37) {
		tmp = sqrt(x) / (x * 3.0);
	} else {
		tmp = sqrt(x) * ((y * 3.0) + -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 6d-37) then
        tmp = sqrt(x) / (x * 3.0d0)
    else
        tmp = sqrt(x) * ((y * 3.0d0) + (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6e-37) {
		tmp = Math.sqrt(x) / (x * 3.0);
	} else {
		tmp = Math.sqrt(x) * ((y * 3.0) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6e-37:
		tmp = math.sqrt(x) / (x * 3.0)
	else:
		tmp = math.sqrt(x) * ((y * 3.0) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6e-37)
		tmp = Float64(sqrt(x) / Float64(x * 3.0));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(y * 3.0) + -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6e-37)
		tmp = sqrt(x) / (x * 3.0);
	else
		tmp = sqrt(x) * ((y * 3.0) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6e-37], N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(y * 3.0), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6e-37

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 98.4%

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

      3. +-commutative 98.4%

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

      4. associate--l+ 98.4%

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

      5. +-commutative 98.4%

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

      6. distribute-rgt-in 98.4%

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

      7. *-commutative 98.4%

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

      8. fma-def 98.4%

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

      9. sub-neg 98.4%

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

      10. metadata-eval 98.4%

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

      11. associate-*l/ 98.5%

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

      12. metadata-eval 98.5%

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

      13. *-commutative 98.5%

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

      14. associate-/r* 98.3%

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

      15. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in x around 0 86.2%

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

      1. expm1-log1p-u 80.2%

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

      2. expm1-udef 80.2%

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

      3. clear-num 80.2%

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

      4. un-div-inv 80.2%

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

      5. div-inv 80.2%

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

      6. metadata-eval 80.2%

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

   6. Applied egg-rr 80.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x}}{x \cdot 3}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 80.2%

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

      2. expm1-log1p 86.4%

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

   8. Simplified 86.4%

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

   if 6e-37 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.5%

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

      3. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-rgt-in 99.5%

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

      7. *-commutative 99.5%

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

      8. fma-def 99.6%

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

      9. sub-neg 99.6%

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

      10. metadata-eval 99.6%

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

      11. associate-*l/ 99.6%

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

      12. metadata-eval 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-/r* 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 92.7%

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

      1. sub-neg 92.7%

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

      2. metadata-eval 92.7%

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

      3. distribute-lft-in 92.7%

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

      4. metadata-eval 92.7%

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

   6. Simplified 92.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-37}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 + -3\right)\\ \end{array} \]

Alternative 10: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.75 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.75e-41) (/ (sqrt x) (* x 3.0)) (* (sqrt (* x 9.0)) (+ y -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.75e-41) {
		tmp = sqrt(x) / (x * 3.0);
	} else {
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.75d-41) then
        tmp = sqrt(x) / (x * 3.0d0)
    else
        tmp = sqrt((x * 9.0d0)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.75e-41) {
		tmp = Math.sqrt(x) / (x * 3.0);
	} else {
		tmp = Math.sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.75e-41:
		tmp = math.sqrt(x) / (x * 3.0)
	else:
		tmp = math.sqrt((x * 9.0)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.75e-41)
		tmp = Float64(sqrt(x) / Float64(x * 3.0));
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.75e-41)
		tmp = sqrt(x) / (x * 3.0);
	else
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.75e-41], N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.75000000000000024e-41

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 98.4%

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

      3. +-commutative 98.4%

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

      4. associate--l+ 98.4%

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

      5. +-commutative 98.4%

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

      6. distribute-rgt-in 98.4%

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

      7. *-commutative 98.4%

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

      8. fma-def 98.4%

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

      9. sub-neg 98.4%

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

      10. metadata-eval 98.4%

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

      11. associate-*l/ 98.5%

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

      12. metadata-eval 98.5%

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

      13. *-commutative 98.5%

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

      14. associate-/r* 98.3%

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

      15. metadata-eval 98.3%

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

   3. Simplified 98.3%

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

   4. Taylor expanded in x around 0 86.2%

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

      1. expm1-log1p-u 80.2%

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

      2. expm1-udef 80.2%

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

      3. clear-num 80.2%

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

      4. un-div-inv 80.2%

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

      5. div-inv 80.2%

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

      6. metadata-eval 80.2%

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

   6. Applied egg-rr 80.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{x}}{x \cdot 3}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 80.2%

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

      2. expm1-log1p 86.4%

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

   8. Simplified 86.4%

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

   if 3.75000000000000024e-41 < x

   1. Initial program 99.6%

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

      1. 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. sub-neg 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.8%

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

      4. pow1/2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow1/2 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around inf 92.9%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.75 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 11: 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 99.5%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
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. sub-neg 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. metadata-eval 99.4%

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

   3. sqrt-prod 99.6%

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

   4. pow1/2 99.6%

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

5. Applied egg-rr 99.6%

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

   1. unpow1/2 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in x around inf 60.1%

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

9. Taylor expanded in y around 0 27.5%

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

    1. *-commutative 27.5%

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

11. Simplified 27.5%

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

    1. expm1-log1p-u 0.9%

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

    2. expm1-udef 1.3%

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

    3. add-sqr-sqrt 0.0%

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

    4. sqrt-unprod 2.4%

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

    5. swap-sqr 2.4%

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

    6. add-sqr-sqrt 2.4%

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

    7. metadata-eval 2.4%

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

13. Applied egg-rr 2.4%

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

    1. expm1-def 3.2%

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

    2. expm1-log1p 3.2%

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

15. Simplified 3.2%

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

16. Final simplification 3.2%

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

Alternative 12: 25.6% 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 99.5%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
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. sub-neg 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. metadata-eval 99.4%

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

   3. sqrt-prod 99.6%

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

   4. pow1/2 99.6%

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

5. Applied egg-rr 99.6%

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

   1. unpow1/2 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in x around inf 60.1%

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

9. Taylor expanded in y around 0 27.5%

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

    1. *-commutative 27.5%

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

11. Simplified 27.5%

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

12. Final simplification 27.5%

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

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 2023275 
(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)))