Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D

Percentage Accurate: 99.7% → 99.7%

Time: 10.2s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l- 99.6%

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

   2. +-commutative 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-/r* 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+83} \lor \neg \left(y \leq 1.3 \cdot 10^{+36}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.65e+83) (not (<= y 1.3e+36)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.65e+83) || !(y <= 1.3e+36)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.65d+83)) .or. (.not. (y <= 1.3d+36))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.65e+83) || !(y <= 1.3e+36)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.65e+83) or not (y <= 1.3e+36):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.65e+83) || !(y <= 1.3e+36))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.65e+83) || ~((y <= 1.3e+36)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.65e+83], N[Not[LessEqual[y, 1.3e+36]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.64999999999999992e83 or 1.3000000000000001e36 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-neg-frac 99.5%

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

      6. neg-mul-1 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-*r/ 99.4%

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

      9. fma-def 99.4%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 94.9%

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

      1. pow1 83.0%

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

      2. sqrt-div 83.1%

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

      3. metadata-eval 83.1%

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

   6. Applied egg-rr 95.0%

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

      1. unpow1 83.1%

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

      2. associate-*r/ 83.0%

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

      3. *-rgt-identity 83.0%

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

   8. Simplified 94.9%

      \[\leadsto 1 + -0.3333333333333333 \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]

   if -1.64999999999999992e83 < y < 1.3000000000000001e36

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 95.9%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+83} \lor \neg \left(y \leq 1.3 \cdot 10^{+36}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]

Alternative 3: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+83} \lor \neg \left(y \leq 1.75 \cdot 10^{+34}\right):\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.7e+83) (not (<= y 1.75e+34)))
   (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x)))
   (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.7e+83) || !(y <= 1.75e+34)) {
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.7d+83)) .or. (.not. (y <= 1.75d+34))) then
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) / sqrt(x))
    else
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.7e+83) || !(y <= 1.75e+34)) {
		tmp = 1.0 + ((y * -0.3333333333333333) / Math.sqrt(x));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.7e+83) or not (y <= 1.75e+34):
		tmp = 1.0 + ((y * -0.3333333333333333) / math.sqrt(x))
	else:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.7e+83) || !(y <= 1.75e+34))
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.7e+83) || ~((y <= 1.75e+34)))
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	else
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.7e+83], N[Not[LessEqual[y, 1.75e+34]], $MachinePrecision]], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6999999999999999e83 or 1.74999999999999999e34 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. distribute-neg-in 99.5%

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

      5. distribute-neg-frac 99.5%

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

      6. neg-mul-1 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-*r/ 99.4%

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

      9. fma-def 99.4%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 94.9%

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

      1. pow1 83.0%

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

      2. sqrt-div 83.1%

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

      3. metadata-eval 83.1%

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

   6. Applied egg-rr 95.0%

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

      1. unpow1 83.1%

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

      2. *-commutative 83.1%

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

      3. associate-*r/ 83.0%

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

      4. *-rgt-identity 83.0%

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

      5. associate-*l/ 83.1%

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

   8. Simplified 95.1%

      \[\leadsto 1 + \color{blue}{\frac{y \cdot -0.3333333333333333}{\sqrt{x}}} \]

   if -1.6999999999999999e83 < y < 1.74999999999999999e34

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 95.9%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+83} \lor \neg \left(y \leq 1.75 \cdot 10^{+34}\right):\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]

Alternative 4: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+83}:\\ \;\;\;\;1 + \left(y \cdot -0.3333333333333333\right) \cdot \frac{1}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+36}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e+83)
   (+ 1.0 (* (* y -0.3333333333333333) (/ 1.0 (sqrt x))))
   (if (<= y 9.5e+36)
     (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))
     (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e+83) {
		tmp = 1.0 + ((y * -0.3333333333333333) * (1.0 / sqrt(x)));
	} else if (y <= 9.5e+36) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.65d+83)) then
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) * (1.0d0 / sqrt(x)))
    else if (y <= 9.5d+36) then
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    else
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.65e+83) {
		tmp = 1.0 + ((y * -0.3333333333333333) * (1.0 / Math.sqrt(x)));
	} else if (y <= 9.5e+36) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = 1.0 + ((y * -0.3333333333333333) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.65e+83:
		tmp = 1.0 + ((y * -0.3333333333333333) * (1.0 / math.sqrt(x)))
	elif y <= 9.5e+36:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	else:
		tmp = 1.0 + ((y * -0.3333333333333333) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.65e+83)
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) * Float64(1.0 / sqrt(x))));
	elseif (y <= 9.5e+36)
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	else
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.65e+83)
		tmp = 1.0 + ((y * -0.3333333333333333) * (1.0 / sqrt(x)));
	elseif (y <= 9.5e+36)
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	else
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.65e+83], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[(1.0 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e+36], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.64999999999999992e83

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. sub-neg 99.4%

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

      3. +-commutative 99.4%

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

      4. distribute-neg-in 99.4%

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

      5. distribute-neg-frac 99.4%

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

      6. neg-mul-1 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-*r/ 99.4%

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

      9. fma-def 99.3%

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

      10. associate-/r* 99.5%

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

      11. metadata-eval 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.5%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 96.5%

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

      1. pow1 92.6%

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

      2. sqrt-div 92.5%

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

      3. metadata-eval 92.5%

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

   6. Applied egg-rr 96.5%

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

      1. unpow1 96.5%

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

      2. associate-*r* 96.7%

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

   8. Simplified 96.7%

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

   if -1.64999999999999992e83 < y < 9.49999999999999974e36

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 95.9%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]

   if 9.49999999999999974e36 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. neg-mul-1 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-*r/ 99.5%

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

      9. fma-def 99.5%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 93.5%

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

      1. pow1 74.3%

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

      2. sqrt-div 74.4%

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

      3. metadata-eval 74.4%

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

   6. Applied egg-rr 93.6%

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

      1. unpow1 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-*r/ 74.3%

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

      4. *-rgt-identity 74.3%

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

      5. associate-*l/ 74.4%

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

   8. Simplified 93.6%

      \[\leadsto 1 + \color{blue}{\frac{y \cdot -0.3333333333333333}{\sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+83}:\\ \;\;\;\;1 + \left(y \cdot -0.3333333333333333\right) \cdot \frac{1}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+36}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 5: 94.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+83}:\\ \;\;\;\;1 - y \cdot \left(0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+36}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e+83)
   (- 1.0 (* y (* 0.3333333333333333 (sqrt (/ 1.0 x)))))
   (if (<= y 5.2e+36)
     (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))
     (+ 1.0 (/ (* y -0.3333333333333333) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e+83) {
		tmp = 1.0 - (y * (0.3333333333333333 * sqrt((1.0 / x))));
	} else if (y <= 5.2e+36) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.65d+83)) then
        tmp = 1.0d0 - (y * (0.3333333333333333d0 * sqrt((1.0d0 / x))))
    else if (y <= 5.2d+36) then
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    else
        tmp = 1.0d0 + ((y * (-0.3333333333333333d0)) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.65e+83) {
		tmp = 1.0 - (y * (0.3333333333333333 * Math.sqrt((1.0 / x))));
	} else if (y <= 5.2e+36) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = 1.0 + ((y * -0.3333333333333333) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.65e+83:
		tmp = 1.0 - (y * (0.3333333333333333 * math.sqrt((1.0 / x))))
	elif y <= 5.2e+36:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	else:
		tmp = 1.0 + ((y * -0.3333333333333333) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.65e+83)
		tmp = Float64(1.0 - Float64(y * Float64(0.3333333333333333 * sqrt(Float64(1.0 / x)))));
	elseif (y <= 5.2e+36)
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	else
		tmp = Float64(1.0 + Float64(Float64(y * -0.3333333333333333) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.65e+83)
		tmp = 1.0 - (y * (0.3333333333333333 * sqrt((1.0 / x))));
	elseif (y <= 5.2e+36)
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	else
		tmp = 1.0 + ((y * -0.3333333333333333) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.65e+83], N[(1.0 - N[(y * N[(0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e+36], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.64999999999999992e83

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in y around inf 96.5%

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

      1. associate-*r* 96.6%

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

      2. *-commutative 96.6%

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

      3. associate-*l* 96.7%

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

   7. Simplified 96.7%

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

   if -1.64999999999999992e83 < y < 5.2000000000000003e36

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 95.9%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]

   if 5.2000000000000003e36 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. distribute-neg-in 99.6%

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

      5. distribute-neg-frac 99.6%

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

      6. neg-mul-1 99.6%

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

      7. *-commutative 99.6%

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

      8. associate-*r/ 99.5%

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

      9. fma-def 99.5%

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

      10. associate-/r* 99.4%

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

      11. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.4%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around inf 93.5%

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

      1. pow1 74.3%

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

      2. sqrt-div 74.4%

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

      3. metadata-eval 74.4%

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

   6. Applied egg-rr 93.6%

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

      1. unpow1 74.4%

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

      2. *-commutative 74.4%

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

      3. associate-*r/ 74.3%

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

      4. *-rgt-identity 74.3%

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

      5. associate-*l/ 74.4%

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

   8. Simplified 93.6%

      \[\leadsto 1 + \color{blue}{\frac{y \cdot -0.3333333333333333}{\sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+83}:\\ \;\;\;\;1 - y \cdot \left(0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+36}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 6: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l- 99.6%

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

   2. +-commutative 99.6%

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

   3. +-commutative 99.6%

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

   4. associate-/r* 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 99.6%

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

5. Final simplification 99.6%

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

Alternative 7: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+106} \lor \neg \left(y \leq 3.35 \cdot 10^{+111}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.2e+106) (not (<= y 3.35e+111)))
   (* -0.3333333333333333 (/ y (sqrt x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.2e+106) || !(y <= 3.35e+111)) {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-3.2d+106)) .or. (.not. (y <= 3.35d+111))) then
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.2e+106) || !(y <= 3.35e+111)) {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.2e+106) or not (y <= 3.35e+111):
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.2e+106) || !(y <= 3.35e+111))
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.2e+106) || ~((y <= 3.35e+111)))
		tmp = -0.3333333333333333 * (y / sqrt(x));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.2e+106], N[Not[LessEqual[y, 3.35e+111]], $MachinePrecision]], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.1999999999999998e106 or 3.3500000000000001e111 < y

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. +-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in y around inf 97.5%

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

      1. pow1 97.5%

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

      2. sqrt-div 97.6%

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

      3. metadata-eval 97.6%

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

   7. Applied egg-rr 97.6%

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

      1. unpow1 97.6%

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

      2. associate-*r/ 97.5%

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

      3. *-rgt-identity 97.5%

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

   9. Simplified 97.5%

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

   if -3.1999999999999998e106 < y < 3.3500000000000001e111

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 91.6%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. div-inv 91.6%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]

      2. clear-num 91.6%

         \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} \]

      3. div-inv 91.6%

         \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      4. metadata-eval 91.6%

         \[\leadsto 1 - \frac{1}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 91.6%

      \[\leadsto 1 - \color{blue}{\frac{1}{x \cdot 9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+106} \lor \neg \left(y \leq 3.35 \cdot 10^{+111}\right):\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 8: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+106}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+109}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.2e+106)
   (* y (/ -0.3333333333333333 (sqrt x)))
   (if (<= y 1.1e+109)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (* -0.3333333333333333 (/ y (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+106) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else if (y <= 1.1e+109) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.3333333333333333 * (y / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.2d+106)) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else if (y <= 1.1d+109) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (-0.3333333333333333d0) * (y / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+106) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else if (y <= 1.1e+109) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = -0.3333333333333333 * (y / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.2e+106:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	elif y <= 1.1e+109:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = -0.3333333333333333 * (y / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.2e+106)
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	elseif (y <= 1.1e+109)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(-0.3333333333333333 * Float64(y / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.2e+106)
		tmp = y * (-0.3333333333333333 / sqrt(x));
	elseif (y <= 1.1e+109)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = -0.3333333333333333 * (y / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.2e+106], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+109], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1999999999999998e106

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. div-inv 99.6%

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

      2. div-inv 99.4%

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

      3. metadata-eval 99.4%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \left(y \cdot \color{blue}{0.3333333333333333}\right) \cdot \frac{1}{\sqrt{x}}\right) \]

      4. pow1/2 99.4%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \left(y \cdot 0.3333333333333333\right) \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

      5. pow-flip 99.3%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \left(y \cdot 0.3333333333333333\right) \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

      6. metadata-eval 99.3%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \left(y \cdot 0.3333333333333333\right) \cdot {x}^{\color{blue}{-0.5}}\right) \]

   5. Applied egg-rr 99.3%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\left(y \cdot 0.3333333333333333\right) \cdot {x}^{-0.5}}\right) \]
   6. Step-by-step derivation

      1. *-commutative 99.3%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)}\right) \]

   7. Simplified 99.3%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)}\right) \]
   8. Step-by-step derivation

      1. metadata-eval 99.3%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + {x}^{-0.5} \cdot \left(y \cdot \color{blue}{\frac{1}{3}}\right)\right) \]

      2. div-inv 99.5%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + {x}^{-0.5} \cdot \color{blue}{\frac{y}{3}}\right) \]

   9. Applied egg-rr 99.5%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + {x}^{-0.5} \cdot \color{blue}{\frac{y}{3}}\right) \]

   10. Taylor expanded in y around inf 96.6%

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

       1. associate-*r* 96.6%

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

       2. *-commutative 96.6%

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

       3. associate-*l* 96.8%

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

   12. Simplified 96.8%

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

       1. expm1-log1p-u 47.6%

          \[\leadsto y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\right)} \]

       2. expm1-udef 4.9%

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

       3. sqrt-div 4.9%

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

       4. metadata-eval 4.9%

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

       5. un-div-inv 4.9%

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

   14. Applied egg-rr 4.9%

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

       1. expm1-def 47.6%

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

       2. expm1-log1p 96.8%

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

   16. Simplified 96.8%

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

   if -3.1999999999999998e106 < y < 1.1e109

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 91.6%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. div-inv 91.6%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]

      2. clear-num 91.6%

         \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} \]

      3. div-inv 91.6%

         \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      4. metadata-eval 91.6%

         \[\leadsto 1 - \frac{1}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 91.6%

      \[\leadsto 1 - \color{blue}{\frac{1}{x \cdot 9}} \]

   if 1.1e109 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in y around inf 98.6%

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

      1. pow1 98.6%

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

      2. sqrt-div 98.9%

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

      3. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. unpow1 98.9%

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

      2. associate-*r/ 98.7%

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

      3. *-rgt-identity 98.7%

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

   9. Simplified 98.7%

      \[\leadsto -0.3333333333333333 \cdot \color{blue}{\frac{y}{\sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+106}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+109}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \end{array} \]

Alternative 9: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+106}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+109}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.2e+106)
   (* y (/ -0.3333333333333333 (sqrt x)))
   (if (<= y 1.1e+109)
     (+ 1.0 (/ -1.0 (* x 9.0)))
     (/ (* y -0.3333333333333333) (sqrt x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+106) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else if (y <= 1.1e+109) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) / sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.2d+106)) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else if (y <= 1.1d+109) then
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    else
        tmp = (y * (-0.3333333333333333d0)) / sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+106) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else if (y <= 1.1e+109) {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	} else {
		tmp = (y * -0.3333333333333333) / Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.2e+106:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	elif y <= 1.1e+109:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	else:
		tmp = (y * -0.3333333333333333) / math.sqrt(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.2e+106)
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	elseif (y <= 1.1e+109)
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	else
		tmp = Float64(Float64(y * -0.3333333333333333) / sqrt(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.2e+106)
		tmp = y * (-0.3333333333333333 / sqrt(x));
	elseif (y <= 1.1e+109)
		tmp = 1.0 + (-1.0 / (x * 9.0));
	else
		tmp = (y * -0.3333333333333333) / sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.2e+106], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+109], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * -0.3333333333333333), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1999999999999998e106

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. div-inv 99.6%

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

      2. div-inv 99.4%

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

      3. metadata-eval 99.4%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \left(y \cdot \color{blue}{0.3333333333333333}\right) \cdot \frac{1}{\sqrt{x}}\right) \]

      4. pow1/2 99.4%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \left(y \cdot 0.3333333333333333\right) \cdot \frac{1}{\color{blue}{{x}^{0.5}}}\right) \]

      5. pow-flip 99.3%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \left(y \cdot 0.3333333333333333\right) \cdot \color{blue}{{x}^{\left(-0.5\right)}}\right) \]

      6. metadata-eval 99.3%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \left(y \cdot 0.3333333333333333\right) \cdot {x}^{\color{blue}{-0.5}}\right) \]

   5. Applied egg-rr 99.3%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{\left(y \cdot 0.3333333333333333\right) \cdot {x}^{-0.5}}\right) \]
   6. Step-by-step derivation

      1. *-commutative 99.3%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)}\right) \]

   7. Simplified 99.3%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + \color{blue}{{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)}\right) \]
   8. Step-by-step derivation

      1. metadata-eval 99.3%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + {x}^{-0.5} \cdot \left(y \cdot \color{blue}{\frac{1}{3}}\right)\right) \]

      2. div-inv 99.5%

         \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + {x}^{-0.5} \cdot \color{blue}{\frac{y}{3}}\right) \]

   9. Applied egg-rr 99.5%

      \[\leadsto 1 - \left(\frac{1}{x \cdot 9} + {x}^{-0.5} \cdot \color{blue}{\frac{y}{3}}\right) \]

   10. Taylor expanded in y around inf 96.6%

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

       1. associate-*r* 96.6%

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

       2. *-commutative 96.6%

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

       3. associate-*l* 96.8%

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

   12. Simplified 96.8%

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

       1. expm1-log1p-u 47.6%

          \[\leadsto y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\right)\right)} \]

       2. expm1-udef 4.9%

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

       3. sqrt-div 4.9%

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

       4. metadata-eval 4.9%

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

       5. un-div-inv 4.9%

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

   14. Applied egg-rr 4.9%

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

       1. expm1-def 47.6%

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

       2. expm1-log1p 96.8%

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

   16. Simplified 96.8%

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

   if -3.1999999999999998e106 < y < 1.1e109

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 91.6%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. div-inv 91.6%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]

      2. clear-num 91.6%

         \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} \]

      3. div-inv 91.6%

         \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      4. metadata-eval 91.6%

         \[\leadsto 1 - \frac{1}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 91.6%

      \[\leadsto 1 - \color{blue}{\frac{1}{x \cdot 9}} \]

   if 1.1e109 < y

   1. Initial program 99.6%

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

      1. associate--l- 99.6%

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

      2. +-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-/r* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.6%

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

   5. Taylor expanded in y around inf 98.6%

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

      1. pow1 98.6%

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

      2. sqrt-div 98.9%

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

      3. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. unpow1 98.9%

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

      2. *-commutative 98.9%

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

      3. associate-*r/ 98.7%

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

      4. *-rgt-identity 98.7%

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

      5. associate-*l/ 98.8%

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

   9. Simplified 98.8%

      \[\leadsto \color{blue}{\frac{y \cdot -0.3333333333333333}{\sqrt{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+106}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+109}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 10: 64.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{1 + \frac{\frac{-0.012345679012345678}{x}}{x}}{1 - \frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e+106)
   (/
    (+ 1.0 (/ (/ -0.012345679012345678 x) x))
    (- 1.0 (/ 0.1111111111111111 x)))
   (+ 1.0 (/ -1.0 (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+106) {
		tmp = (1.0 + ((-0.012345679012345678 / x) / x)) / (1.0 - (0.1111111111111111 / x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9.5d+106)) then
        tmp = (1.0d0 + (((-0.012345679012345678d0) / x) / x)) / (1.0d0 - (0.1111111111111111d0 / x))
    else
        tmp = 1.0d0 + ((-1.0d0) / (x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+106) {
		tmp = (1.0 + ((-0.012345679012345678 / x) / x)) / (1.0 - (0.1111111111111111 / x));
	} else {
		tmp = 1.0 + (-1.0 / (x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5e+106:
		tmp = (1.0 + ((-0.012345679012345678 / x) / x)) / (1.0 - (0.1111111111111111 / x))
	else:
		tmp = 1.0 + (-1.0 / (x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e+106)
		tmp = Float64(Float64(1.0 + Float64(Float64(-0.012345679012345678 / x) / x)) / Float64(1.0 - Float64(0.1111111111111111 / x)));
	else
		tmp = Float64(1.0 + Float64(-1.0 / Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.5e+106)
		tmp = (1.0 + ((-0.012345679012345678 / x) / x)) / (1.0 - (0.1111111111111111 / x));
	else
		tmp = 1.0 + (-1.0 / (x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e+106], N[(N[(1.0 + N[(N[(-0.012345679012345678 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.4999999999999995e106

   1. Initial program 99.4%

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

      1. associate--l- 99.4%

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

      2. +-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-/r* 99.7%

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

   3. Simplified 99.7%

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

      1. associate--r+ 99.7%

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

      2. associate-/r* 99.4%

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

      3. inv-pow 99.4%

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

      4. *-commutative 99.4%

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

      5. unpow-prod-down 99.4%

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

      6. metadata-eval 99.4%

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

      7. inv-pow 99.4%

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

      8. div-inv 99.4%

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

      9. flip-- 82.8%

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

      10. clear-num 82.8%

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

      11. frac-sub 82.7%

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

   5. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{\frac{\left(1 - \frac{0.012345679012345678}{x \cdot x}\right) \cdot \frac{\sqrt{x} \cdot 3}{y} - \left(1 + \frac{-0.1111111111111111}{x}\right) \cdot 1}{\left(1 + \frac{-0.1111111111111111}{x}\right) \cdot \frac{\sqrt{x} \cdot 3}{y}}} \]
   6. Step-by-step derivation

   7. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(3 \cdot \frac{\sqrt{x}}{y}, 1 + \frac{-0.012345679012345678}{x \cdot x}, -1 + \frac{0.1111111111111111}{x}\right)}{\left(\frac{-0.1111111111111111}{x} + 1\right) \cdot \left(3 \cdot \frac{\sqrt{x}}{y}\right)}} \]

   8. Taylor expanded in y around 0 15.1%

      \[\leadsto \color{blue}{\frac{1 - 0.012345679012345678 \cdot \frac{1}{{x}^{2}}}{1 - 0.1111111111111111 \cdot \frac{1}{x}}} \]
   9. Step-by-step derivation

      1. cancel-sign-sub-inv 15.1%

         \[\leadsto \frac{\color{blue}{1 + \left(-0.012345679012345678\right) \cdot \frac{1}{{x}^{2}}}}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]

      2. metadata-eval 15.1%

         \[\leadsto \frac{1 + \color{blue}{-0.012345679012345678} \cdot \frac{1}{{x}^{2}}}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]

      3. associate-*r/ 15.1%

         \[\leadsto \frac{1 + \color{blue}{\frac{-0.012345679012345678 \cdot 1}{{x}^{2}}}}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]

      4. metadata-eval 15.1%

         \[\leadsto \frac{1 + \frac{\color{blue}{-0.012345679012345678}}{{x}^{2}}}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]

      5. unpow2 15.1%

         \[\leadsto \frac{1 + \frac{-0.012345679012345678}{\color{blue}{x \cdot x}}}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]

      6. associate-/r* 15.1%

         \[\leadsto \frac{1 + \color{blue}{\frac{\frac{-0.012345679012345678}{x}}{x}}}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]

      7. associate-*r/ 15.1%

         \[\leadsto \frac{1 + \frac{\frac{-0.012345679012345678}{x}}{x}}{1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}}} \]

      8. metadata-eval 15.1%

         \[\leadsto \frac{1 + \frac{\frac{-0.012345679012345678}{x}}{x}}{1 - \frac{\color{blue}{0.1111111111111111}}{x}} \]

   10. Simplified 15.1%

       \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.012345679012345678}{x}}{x}}{1 - \frac{0.1111111111111111}{x}}} \]

   if -9.4999999999999995e106 < y

   1. Initial program 99.7%

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

      1. associate--l- 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. distribute-neg-in 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. neg-mul-1 99.7%

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

      7. *-commutative 99.7%

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

      8. associate-*r/ 99.6%

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

      9. fma-def 99.6%

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

      10. associate-/r* 99.6%

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

      11. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.6%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in y around 0 78.4%

      \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
   5. Step-by-step derivation

      1. div-inv 78.4%

         \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]

      2. clear-num 78.4%

         \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} \]

      3. div-inv 78.4%

         \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      4. metadata-eval 78.4%

         \[\leadsto 1 - \frac{1}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 78.4%

      \[\leadsto 1 - \color{blue}{\frac{1}{x \cdot 9}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{1 + \frac{\frac{-0.012345679012345678}{x}}{x}}{1 - \frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x \cdot 9}\\ \end{array} \]

Alternative 11: 60.8% accurate, 16.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{1}{x} \cdot -0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112) (* (/ 1.0 x) -0.1111111111111111) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = (1.0 / x) * -0.1111111111111111;
	} else {
		tmp = 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 = (1.0d0 / x) * (-0.1111111111111111d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = (1.0 / x) * -0.1111111111111111;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = (1.0 / x) * -0.1111111111111111
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(Float64(1.0 / x) * -0.1111111111111111);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = (1.0 / x) * -0.1111111111111111;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(N[(1.0 / x), $MachinePrecision] * -0.1111111111111111), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. +-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-/r* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 99.4%

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

   5. Taylor expanded in x around 0 65.9%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} \]
   6. Step-by-step derivation

      1. clear-num 65.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{x}{-0.1111111111111111}}} \]

      2. inv-pow 65.8%

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

   7. Applied egg-rr 65.8%

      \[\leadsto \color{blue}{{\left(\frac{x}{-0.1111111111111111}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. div-inv 65.9%

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

      2. unpow-prod-down 65.9%

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

      3. inv-pow 65.9%

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

      4. metadata-eval 65.9%

         \[\leadsto \frac{1}{x} \cdot {\color{blue}{-9}}^{-1} \]

      5. metadata-eval 65.9%

         \[\leadsto \frac{1}{x} \cdot \color{blue}{-0.1111111111111111} \]

   9. Applied egg-rr 65.9%

      \[\leadsto \color{blue}{\frac{1}{x} \cdot -0.1111111111111111} \]

   if 0.112000000000000002 < x

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in x around inf 65.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{1}{x} \cdot -0.1111111111111111\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12: 61.9% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + 0.1111111111111111 \cdot \frac{-1}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0 + (0.1111111111111111 * (-1.0 / x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l- 99.6%

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

   2. sub-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-neg-in 99.6%

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

   5. distribute-neg-frac 99.6%

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

   6. neg-mul-1 99.6%

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

   7. *-commutative 99.6%

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

   8. associate-*r/ 99.6%

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

   9. fma-def 99.6%

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

   10. associate-/r* 99.6%

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

   11. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   12. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   13. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   14. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   15. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

4. Taylor expanded in y around 0 66.2%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]

5. Final simplification 66.2%

   \[\leadsto 1 + 0.1111111111111111 \cdot \frac{-1}{x} \]

Alternative 13: 61.9% accurate, 16.1× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{-1}{x \cdot 9} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l- 99.6%

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

   2. sub-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-neg-in 99.6%

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

   5. distribute-neg-frac 99.6%

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

   6. neg-mul-1 99.6%

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

   7. *-commutative 99.6%

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

   8. associate-*r/ 99.6%

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

   9. fma-def 99.6%

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

   10. associate-/r* 99.6%

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

   11. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   12. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   13. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   14. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   15. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

4. Taylor expanded in y around 0 66.2%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
5. Step-by-step derivation

   1. div-inv 66.1%

      \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111}{x}} \]

   2. clear-num 66.1%

      \[\leadsto 1 - \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} \]

   3. div-inv 66.2%

      \[\leadsto 1 - \frac{1}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

   4. metadata-eval 66.2%

      \[\leadsto 1 - \frac{1}{x \cdot \color{blue}{9}} \]

6. Applied egg-rr 66.2%

   \[\leadsto 1 - \color{blue}{\frac{1}{x \cdot 9}} \]

7. Final simplification 66.2%

   \[\leadsto 1 + \frac{-1}{x \cdot 9} \]

Alternative 14: 60.8% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 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 = (-0.1111111111111111d0) / x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(-0.1111111111111111 / x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.5%

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

      1. associate--l- 99.5%

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

      2. +-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-/r* 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 99.4%

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

   5. Taylor expanded in x around 0 65.9%

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

   if 0.112000000000000002 < x

   1. Initial program 99.8%

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

      1. associate--l- 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. distribute-neg-frac 99.8%

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

      6. neg-mul-1 99.8%

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

      7. *-commutative 99.8%

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

      8. associate-*r/ 99.7%

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

      9. fma-def 99.7%

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

      10. associate-/r* 99.7%

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

      11. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

      12. *-commutative 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

      13. associate-/r* 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

      14. distribute-neg-frac 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

      15. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

      16. metadata-eval 99.7%

          \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

   4. Taylor expanded in x around inf 65.1%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 15: 61.9% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{0.1111111111111111}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (/ 0.1111111111111111 x)))

C

double code(double x, double y) {
	return 1.0 - (0.1111111111111111 / x);
}

Fortran

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

Java

public static double code(double x, double y) {
	return 1.0 - (0.1111111111111111 / x);
}

Python

def code(x, y):
	return 1.0 - (0.1111111111111111 / x)

Julia

function code(x, y)
	return Float64(1.0 - Float64(0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - (0.1111111111111111 / x);
end

Wolfram

code[x_, y_] := N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate--l- 99.6%

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

   2. sub-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-neg-in 99.6%

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

   5. distribute-neg-frac 99.6%

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

   6. neg-mul-1 99.6%

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

   7. *-commutative 99.6%

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

   8. associate-*r/ 99.6%

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

   9. fma-def 99.6%

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

   10. associate-/r* 99.6%

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

   11. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   12. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   13. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   14. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   15. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

4. Taylor expanded in y around 0 66.2%

   \[\leadsto \color{blue}{1 - 0.1111111111111111 \cdot \frac{1}{x}} \]
5. Step-by-step derivation

   1. associate-*r/ 66.1%

      \[\leadsto 1 - \color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} \]

   2. metadata-eval 66.1%

      \[\leadsto 1 - \frac{\color{blue}{0.1111111111111111}}{x} \]

6. Simplified 66.1%

   \[\leadsto \color{blue}{1 - \frac{0.1111111111111111}{x}} \]

7. Final simplification 66.1%

   \[\leadsto 1 - \frac{0.1111111111111111}{x} \]

Alternative 16: 30.7% accurate, 113.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.6%

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

   1. associate--l- 99.6%

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

   2. sub-neg 99.6%

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

   3. +-commutative 99.6%

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

   4. distribute-neg-in 99.6%

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

   5. distribute-neg-frac 99.6%

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

   6. neg-mul-1 99.6%

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

   7. *-commutative 99.6%

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

   8. associate-*r/ 99.6%

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

   9. fma-def 99.6%

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

   10. associate-/r* 99.6%

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

   11. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{\color{blue}{-0.3333333333333333}}{\sqrt{x}}, -\frac{1}{x \cdot 9}\right) \]

   12. *-commutative 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\frac{1}{\color{blue}{9 \cdot x}}\right) \]

   13. associate-/r* 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, -\color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   14. distribute-neg-frac 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \color{blue}{\frac{-\frac{1}{9}}{x}}\right) \]

   15. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-\color{blue}{0.1111111111111111}}{x}\right) \]

   16. metadata-eval 99.6%

       \[\leadsto 1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{\color{blue}{-0.1111111111111111}}{x}\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{1 + \mathsf{fma}\left(y, \frac{-0.3333333333333333}{\sqrt{x}}, \frac{-0.1111111111111111}{x}\right)} \]

4. Taylor expanded in x around inf 33.5%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 33.5%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023171 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64

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

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