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

Percentage Accurate: 99.7% → 99.7%

Time: 12.9s

Alternatives: 20

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

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

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]

Derivation

1. Initial program 99.7%

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

2. Final simplification 99.7%

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

Alternative 2: 94.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+56} \lor \neg \left(y \leq 4.7 \cdot 10^{+33}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.8e+56) (not (<= y 4.7e+33)))
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.8e+56) || !(y <= 4.7e+33)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / 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 <= (-3.8d+56)) .or. (.not. (y <= 4.7d+33))) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.8e+56) || !(y <= 4.7e+33)) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.8e+56) or not (y <= 4.7e+33):
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.8e+56) || !(y <= 4.7e+33))
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.8e+56) || ~((y <= 4.7e+33)))
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.8e+56], N[Not[LessEqual[y, 4.7e+33]], $MachinePrecision]], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.79999999999999996e56 or 4.6999999999999998e33 < 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. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 88.9%

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

   if -3.79999999999999996e56 < y < 4.6999999999999998e33

   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. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-/r* 99.7%

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

      8. metadata-eval 99.7%

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

      9. sub-neg 99.7%

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

      10. distribute-frac-neg 99.7%

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

      11. +-commutative 99.7%

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

      12. neg-mul-1 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-*r/ 99.7%

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

      15. fma-def 99.7%

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

      16. associate-/r* 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 99.0%

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

      1. cancel-sign-sub-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. associate-*r/ 99.1%

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

      4. metadata-eval 99.1%

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

   8. Simplified 99.1%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+56} \lor \neg \left(y \leq 4.7 \cdot 10^{+33}\right):\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 3: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+56} \lor \neg \left(y \leq 6.2 \cdot 10^{+33}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.9e+56) (not (<= y 6.2e+33)))
   (- 1.0 (/ y (sqrt (* x 9.0))))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.9e+56) || !(y <= 6.2e+33)) {
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / 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 <= (-3.9d+56)) .or. (.not. (y <= 6.2d+33))) then
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3.9e+56) || !(y <= 6.2e+33)) {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3.9e+56) or not (y <= 6.2e+33):
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3.9e+56) || !(y <= 6.2e+33))
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3.9e+56) || ~((y <= 6.2e+33)))
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3.9e+56], N[Not[LessEqual[y, 6.2e+33]], $MachinePrecision]], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.89999999999999994e56 or 6.2e33 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

   3. Taylor expanded in x around inf 89.0%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

      1. *-commutative 89.0%

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

      2. metadata-eval 89.0%

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

      3. sqrt-prod 89.0%

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

      4. pow1/2 89.0%

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

   5. Applied egg-rr 89.0%

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

      1. unpow1/2 89.0%

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

   7. Simplified 89.0%

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

   if -3.89999999999999994e56 < y < 6.2e33

   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. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-/r* 99.7%

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

      8. metadata-eval 99.7%

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

      9. sub-neg 99.7%

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

      10. distribute-frac-neg 99.7%

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

      11. +-commutative 99.7%

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

      12. neg-mul-1 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-*r/ 99.7%

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

      15. fma-def 99.7%

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

      16. associate-/r* 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 99.0%

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

      1. cancel-sign-sub-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. associate-*r/ 99.1%

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

      4. metadata-eval 99.1%

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

   8. Simplified 99.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+56} \lor \neg \left(y \leq 6.2 \cdot 10^{+33}\right):\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 4: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+56}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+33}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.6e+56)
   (+ 1.0 (* -0.3333333333333333 (/ y (sqrt x))))
   (if (<= y 2e+33)
     (+ 1.0 (/ -0.1111111111111111 x))
     (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+56) {
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	} else if (y <= 2e+33) {
		tmp = 1.0 + (-0.1111111111111111 / 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 <= (-8.6d+56)) then
        tmp = 1.0d0 + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else if (y <= 2d+33) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / 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 <= -8.6e+56) {
		tmp = 1.0 + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else if (y <= 2e+33) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 + (y * (-0.3333333333333333 / Math.sqrt(x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.6e+56:
		tmp = 1.0 + (-0.3333333333333333 * (y / math.sqrt(x)))
	elif y <= 2e+33:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 + (y * (-0.3333333333333333 / math.sqrt(x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.6e+56)
		tmp = Float64(1.0 + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	elseif (y <= 2e+33)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.6e+56)
		tmp = 1.0 + (-0.3333333333333333 * (y / sqrt(x)));
	elseif (y <= 2e+33)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 + (y * (-0.3333333333333333 / sqrt(x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.6e+56], N[(1.0 + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+33], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.6000000000000007e56

   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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 92.0%

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

   if -8.6000000000000007e56 < y < 1.9999999999999999e33

   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. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-/r* 99.7%

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

      8. metadata-eval 99.7%

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

      9. sub-neg 99.7%

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

      10. distribute-frac-neg 99.7%

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

      11. +-commutative 99.7%

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

      12. neg-mul-1 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-*r/ 99.7%

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

      15. fma-def 99.7%

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

      16. associate-/r* 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 99.0%

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

      1. cancel-sign-sub-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. associate-*r/ 99.1%

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

      4. metadata-eval 99.1%

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

   8. Simplified 99.1%

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

   if 1.9999999999999999e33 < 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. sub-neg 99.6%

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

      2. distribute-frac-neg 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.3%

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

      8. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around inf 86.2%

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

      1. *-commutative 86.2%

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

      2. associate-*l/ 86.2%

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

      3. associate-*r/ 86.3%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 11.3%

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

      6. frac-times 11.3%

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

      7. metadata-eval 11.3%

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

      8. add-sqr-sqrt 11.3%

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

      9. clear-num 11.3%

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

      10. inv-pow 11.3%

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

      11. div-inv 11.3%

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

      12. metadata-eval 11.3%

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

      13. inv-pow 11.3%

          \[\leadsto 1 + y \cdot \sqrt{\color{blue}{\frac{1}{x \cdot 9}}} \]

      14. sqrt-div 11.3%

          \[\leadsto 1 + y \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{x \cdot 9}}} \]

      15. metadata-eval 11.3%

          \[\leadsto 1 + y \cdot \frac{\color{blue}{1}}{\sqrt{x \cdot 9}} \]

      16. sqrt-prod 11.3%

          \[\leadsto 1 + y \cdot \frac{1}{\color{blue}{\sqrt{x} \cdot \sqrt{9}}} \]

      17. metadata-eval 11.3%

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

      18. *-commutative 11.3%

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

      19. div-inv 11.3%

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

      20. expm1-log1p-u 11.3%

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

      21. expm1-udef 11.3%

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

   6. Applied egg-rr 13.1%

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

      1. expm1-def 13.1%

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

      2. expm1-log1p 86.3%

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

   8. Simplified 86.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+56}:\\ \;\;\;\;1 + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+33}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \end{array} \]

Alternative 5: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+57}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+33}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.06e+57)
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (if (<= y 2.7e+33)
     (+ 1.0 (/ -0.1111111111111111 x))
     (- 1.0 (/ y (sqrt (* x 9.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+57) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else if (y <= 2.7e+33) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 - (y / sqrt((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 <= (-1.06d+57)) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else if (y <= 2.7d+33) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.06e+57) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else if (y <= 2.7e+33) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.06e+57:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	elif y <= 2.7e+33:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.06e+57)
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	elseif (y <= 2.7e+33)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.06e+57)
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	elseif (y <= 2.7e+33)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.06e+57], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+33], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.06e57

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.6%

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

   3. Taylor expanded in x around inf 92.0%

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

   if -1.06e57 < y < 2.69999999999999991e33

   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. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-/r* 99.7%

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

      8. metadata-eval 99.7%

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

      9. sub-neg 99.7%

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

      10. distribute-frac-neg 99.7%

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

      11. +-commutative 99.7%

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

      12. neg-mul-1 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-*r/ 99.7%

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

      15. fma-def 99.7%

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

      16. associate-/r* 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 99.0%

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

      1. cancel-sign-sub-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. associate-*r/ 99.1%

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

      4. metadata-eval 99.1%

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

   8. Simplified 99.1%

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

   if 2.69999999999999991e33 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

   3. Taylor expanded in x around inf 86.4%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

      1. *-commutative 86.4%

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

      2. metadata-eval 86.4%

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

      3. sqrt-prod 86.4%

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

      4. pow1/2 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. unpow1/2 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+57}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+33}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]

Alternative 6: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+57}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+33}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{3}}{\sqrt{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.26e+57)
   (- 1.0 (/ y (* 3.0 (sqrt x))))
   (if (<= y 1.25e+33)
     (+ 1.0 (/ -0.1111111111111111 x))
     (- 1.0 (/ (/ y 3.0) (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.26e+57) {
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	} else if (y <= 1.25e+33) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 - ((y / 3.0) / 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 <= (-1.26d+57)) then
        tmp = 1.0d0 - (y / (3.0d0 * sqrt(x)))
    else if (y <= 1.25d+33) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = 1.0d0 - ((y / 3.0d0) / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.26e+57) {
		tmp = 1.0 - (y / (3.0 * Math.sqrt(x)));
	} else if (y <= 1.25e+33) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = 1.0 - ((y / 3.0) / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.26e+57:
		tmp = 1.0 - (y / (3.0 * math.sqrt(x)))
	elif y <= 1.25e+33:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = 1.0 - ((y / 3.0) / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.26e+57)
		tmp = Float64(1.0 - Float64(y / Float64(3.0 * sqrt(x))));
	elseif (y <= 1.25e+33)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(1.0 - Float64(Float64(y / 3.0) / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.26e+57)
		tmp = 1.0 - (y / (3.0 * sqrt(x)));
	elseif (y <= 1.25e+33)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = 1.0 - ((y / 3.0) / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.26e+57], N[(1.0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+33], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / 3.0), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.26e57

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0 99.6%

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

   3. Taylor expanded in x around inf 92.0%

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

   if -1.26e57 < y < 1.24999999999999993e33

   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. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-/r* 99.7%

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

      8. metadata-eval 99.7%

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

      9. sub-neg 99.7%

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

      10. distribute-frac-neg 99.7%

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

      11. +-commutative 99.7%

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

      12. neg-mul-1 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-*r/ 99.7%

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

      15. fma-def 99.7%

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

      16. associate-/r* 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 99.0%

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

      1. cancel-sign-sub-inv 99.0%

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

      2. metadata-eval 99.0%

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

      3. associate-*r/ 99.1%

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

      4. metadata-eval 99.1%

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

   8. Simplified 99.1%

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

   if 1.24999999999999993e33 < y

   1. Initial program 99.6%

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

   2. Taylor expanded in x around 0 99.6%

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

   3. Taylor expanded in x around inf 86.4%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

      1. sub-neg 86.4%

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

      2. associate-/r* 86.5%

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

   5. Applied egg-rr 86.5%

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

      1. sub-neg 86.5%

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

   7. Simplified 86.5%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{+57}:\\ \;\;\;\;1 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+33}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\frac{y}{3}}{\sqrt{x}}\\ \end{array} \]

Alternative 7: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.039:\\ \;\;\;\;\frac{-0.1111111111111111}{x} + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.039)
   (+ (/ -0.1111111111111111 x) (* -0.3333333333333333 (/ y (sqrt x))))
   (- 1.0 (/ y (sqrt (* x 9.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.039) {
		tmp = (-0.1111111111111111 / x) + (-0.3333333333333333 * (y / sqrt(x)));
	} else {
		tmp = 1.0 - (y / sqrt((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 (x <= 0.039d0) then
        tmp = ((-0.1111111111111111d0) / x) + ((-0.3333333333333333d0) * (y / sqrt(x)))
    else
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.039) {
		tmp = (-0.1111111111111111 / x) + (-0.3333333333333333 * (y / Math.sqrt(x)));
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.039:
		tmp = (-0.1111111111111111 / x) + (-0.3333333333333333 * (y / math.sqrt(x)))
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.039)
		tmp = Float64(Float64(-0.1111111111111111 / x) + Float64(-0.3333333333333333 * Float64(y / sqrt(x))));
	else
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.039)
		tmp = (-0.1111111111111111 / x) + (-0.3333333333333333 * (y / sqrt(x)));
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.039], N[(N[(-0.1111111111111111 / x), $MachinePrecision] + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0389999999999999999

   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. sub-neg 99.7%

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

      2. distribute-frac-neg 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-/r* 99.6%

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

      5. metadata-eval 99.6%

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

      6. neg-mul-1 99.6%

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

      7. times-frac 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 98.4%

      \[\leadsto \color{blue}{\frac{-0.1111111111111111}{x}} + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}} \]

   if 0.0389999999999999999 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

   3. Taylor expanded in x around inf 98.6%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

      1. *-commutative 98.6%

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

      2. metadata-eval 98.6%

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

      3. sqrt-prod 98.6%

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

      4. pow1/2 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. unpow1/2 98.6%

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

   7. Simplified 98.6%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.039:\\ \;\;\;\;\frac{-0.1111111111111111}{x} + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]

Alternative 8: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.039:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.039)
   (- (/ -0.1111111111111111 x) (/ y (* 3.0 (sqrt x))))
   (- 1.0 (/ y (sqrt (* x 9.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.039) {
		tmp = (-0.1111111111111111 / x) - (y / (3.0 * sqrt(x)));
	} else {
		tmp = 1.0 - (y / sqrt((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 (x <= 0.039d0) then
        tmp = ((-0.1111111111111111d0) / x) - (y / (3.0d0 * sqrt(x)))
    else
        tmp = 1.0d0 - (y / sqrt((x * 9.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.039) {
		tmp = (-0.1111111111111111 / x) - (y / (3.0 * Math.sqrt(x)));
	} else {
		tmp = 1.0 - (y / Math.sqrt((x * 9.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.039:
		tmp = (-0.1111111111111111 / x) - (y / (3.0 * math.sqrt(x)))
	else:
		tmp = 1.0 - (y / math.sqrt((x * 9.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.039)
		tmp = Float64(Float64(-0.1111111111111111 / x) - Float64(y / Float64(3.0 * sqrt(x))));
	else
		tmp = Float64(1.0 - Float64(y / sqrt(Float64(x * 9.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.039)
		tmp = (-0.1111111111111111 / x) - (y / (3.0 * sqrt(x)));
	else
		tmp = 1.0 - (y / sqrt((x * 9.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.039], N[(N[(-0.1111111111111111 / x), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0389999999999999999

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.6%

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

   3. Taylor expanded in x around 0 98.5%

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

   if 0.0389999999999999999 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 99.8%

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

   3. Taylor expanded in x around inf 98.6%

      \[\leadsto \color{blue}{1} - \frac{y}{3 \cdot \sqrt{x}} \]
   4. Step-by-step derivation

      1. *-commutative 98.6%

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

      2. metadata-eval 98.6%

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

      3. sqrt-prod 98.6%

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

      4. pow1/2 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. unpow1/2 98.6%

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

   7. Simplified 98.6%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.039:\\ \;\;\;\;\frac{-0.1111111111111111}{x} - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{\sqrt{x \cdot 9}}\\ \end{array} \]

Alternative 9: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- 1.0 (/ 0.1111111111111111 x)) (* -0.3333333333333333 (/ y (sqrt x)))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) + ((-0.3333333333333333d0) * (y / sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / Math.sqrt(x)));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / math.sqrt(x)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) + Float64(-0.3333333333333333 * Float64(y / sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) + (-0.3333333333333333 * (y / sqrt(x)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. sub-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) + -0.3333333333333333 \cdot \frac{y}{\sqrt{x}} \]

Alternative 10: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-0.1111111111111111}{x} + \left(1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (/ -0.1111111111111111 x) (+ 1.0 (* y (/ -0.3333333333333333 (sqrt x))))))

C

double code(double x, double y) {
	return (-0.1111111111111111 / x) + (1.0 + (y * (-0.3333333333333333 / sqrt(x))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((-0.1111111111111111d0) / x) + (1.0d0 + (y * ((-0.3333333333333333d0) / sqrt(x))))
end function

Java

public static double code(double x, double y) {
	return (-0.1111111111111111 / x) + (1.0 + (y * (-0.3333333333333333 / Math.sqrt(x))));
}

Python

def code(x, y):
	return (-0.1111111111111111 / x) + (1.0 + (y * (-0.3333333333333333 / math.sqrt(x))))

Julia

function code(x, y)
	return Float64(Float64(-0.1111111111111111 / x) + Float64(1.0 + Float64(y * Float64(-0.3333333333333333 / sqrt(x)))))
end

MATLAB

function tmp = code(x, y)
	tmp = (-0.1111111111111111 / x) + (1.0 + (y * (-0.3333333333333333 / sqrt(x))));
end

Wolfram

code[x_, y_] := N[(N[(-0.1111111111111111 / x), $MachinePrecision] + N[(1.0 + N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. sub-neg 99.7%

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

   2. +-commutative 99.7%

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

   3. associate--l+ 99.7%

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

   4. distribute-neg-frac 99.7%

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

   5. metadata-eval 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-/r* 99.7%

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

   8. metadata-eval 99.7%

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

   9. sub-neg 99.7%

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

   10. distribute-frac-neg 99.7%

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

   11. +-commutative 99.7%

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

   12. neg-mul-1 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-*r/ 99.6%

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

   15. fma-def 99.6%

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

   16. associate-/r* 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. fma-udef 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

   \[\leadsto \frac{-0.1111111111111111}{x} + \left(1 + y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\right) \]

Alternative 11: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(y / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 99.7%

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

   1. *-commutative 65.8%

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

   2. metadata-eval 65.8%

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

   3. sqrt-prod 65.8%

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

   4. pow1/2 65.8%

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

4. Applied egg-rr 99.7%

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

   1. unpow1/2 65.8%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 12: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

2. Taylor expanded in x around 0 99.7%

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

3. Final simplification 99.7%

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

Alternative 13: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+107} \lor \neg \left(y \leq 1.2 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{-y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4e+107) (not (<= y 1.2e+104)))
   (/ (- y) (* 3.0 (sqrt x)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4e+107) || !(y <= 1.2e+104)) {
		tmp = -y / (3.0 * sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / 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 <= (-4d+107)) .or. (.not. (y <= 1.2d+104))) then
        tmp = -y / (3.0d0 * sqrt(x))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4e+107) || !(y <= 1.2e+104)) {
		tmp = -y / (3.0 * Math.sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4e+107) or not (y <= 1.2e+104):
		tmp = -y / (3.0 * math.sqrt(x))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4e+107) || !(y <= 1.2e+104))
		tmp = Float64(Float64(-y) / Float64(3.0 * sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4e+107) || ~((y <= 1.2e+104)))
		tmp = -y / (3.0 * sqrt(x));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4e+107], N[Not[LessEqual[y, 1.2e+104]], $MachinePrecision]], N[((-y) / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9999999999999999e107 or 1.2e104 < 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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.5%

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

      5. metadata-eval 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.3%

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

      8. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around inf 97.0%

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

   5. Taylor expanded in y around inf 93.4%

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

      1. associate-*r* 93.4%

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

   7. Simplified 93.4%

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

      1. sqrt-div 93.2%

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

      2. metadata-eval 93.2%

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

      3. un-div-inv 93.2%

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

      4. *-commutative 93.2%

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

      5. metadata-eval 93.2%

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

      6. metadata-eval 93.2%

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

      7. distribute-rgt-neg-in 93.2%

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

      8. metadata-eval 93.2%

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

      9. metadata-eval 93.2%

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

      10. div-inv 93.5%

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

      11. distribute-neg-frac 93.5%

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

      12. associate-/l/ 93.5%

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

      13. distribute-neg-frac 93.5%

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

      14. *-commutative 93.5%

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

   9. Applied egg-rr 93.5%

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

   if -3.9999999999999999e107 < y < 1.2e104

   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. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-/r* 99.8%

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

      8. metadata-eval 99.8%

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

      9. sub-neg 99.8%

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

      10. distribute-frac-neg 99.8%

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

      11. +-commutative 99.8%

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

      12. neg-mul-1 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-*r/ 99.7%

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

      15. fma-def 99.7%

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

      16. associate-/r* 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 92.7%

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

      1. cancel-sign-sub-inv 92.7%

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

      2. metadata-eval 92.7%

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

      3. associate-*r/ 92.7%

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

      4. metadata-eval 92.7%

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

   8. Simplified 92.7%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+107} \lor \neg \left(y \leq 1.2 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{-y}{3 \cdot \sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 14: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+107}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+105}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7e+107)
   (* y (/ -0.3333333333333333 (sqrt x)))
   (if (<= y 2.05e+105)
     (+ 1.0 (/ -0.1111111111111111 x))
     (* y (* -0.3333333333333333 (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7e+107) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else if (y <= 2.05e+105) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y * (-0.3333333333333333 * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7d+107)) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else if (y <= 2.05d+105) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = y * ((-0.3333333333333333d0) * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7e+107) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else if (y <= 2.05e+105) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y * (-0.3333333333333333 * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7e+107:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	elif y <= 2.05e+105:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = y * (-0.3333333333333333 * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7e+107)
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	elseif (y <= 2.05e+105)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7e+107)
		tmp = y * (-0.3333333333333333 / sqrt(x));
	elseif (y <= 2.05e+105)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = y * (-0.3333333333333333 * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7e+107], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e+105], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.9999999999999995e107

   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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.5%

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

      5. metadata-eval 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 99.4%

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

   5. Taylor expanded in y around inf 96.0%

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

      1. associate-*r* 96.0%

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

   7. Simplified 96.0%

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

      1. expm1-log1p-u 88.1%

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

      2. expm1-udef 88.1%

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

   9. Applied egg-rr 96.0%

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

       1. +-commutative 96.0%

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

       2. associate--l+ 96.0%

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

       3. metadata-eval 96.0%

          \[\leadsto -0.3333333333333333 \cdot \frac{y}{\sqrt{x}} + \color{blue}{0} \]

       4. +-rgt-identity 96.0%

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

       5. associate-*r/ 95.7%

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

       6. associate-/l* 96.0%

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

       7. associate-/r/ 95.9%

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

   11. Simplified 95.9%

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

   if -6.9999999999999995e107 < y < 2.0500000000000001e105

   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. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-/r* 99.8%

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

      8. metadata-eval 99.8%

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

      9. sub-neg 99.8%

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

      10. distribute-frac-neg 99.8%

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

      11. +-commutative 99.8%

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

      12. neg-mul-1 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-*r/ 99.8%

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

      15. fma-def 99.8%

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

      16. associate-/r* 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 92.2%

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

      1. cancel-sign-sub-inv 92.2%

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

      2. metadata-eval 92.2%

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

      3. associate-*r/ 92.2%

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

      4. metadata-eval 92.2%

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

   8. Simplified 92.2%

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

   if 2.0500000000000001e105 < 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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.5%

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

      5. metadata-eval 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.1%

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

      8. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around inf 96.5%

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

   5. Taylor expanded in y around inf 92.4%

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

      1. associate-*r* 92.4%

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

      2. *-commutative 92.4%

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

      3. associate-*l* 92.6%

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

   7. Simplified 92.6%

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

      1. expm1-log1p-u 45.5%

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

      2. expm1-udef 2.2%

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

      3. inv-pow 2.2%

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

      4. sqrt-pow1 2.2%

         \[\leadsto y \cdot \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 2.2%

         \[\leadsto y \cdot \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   9. Applied egg-rr 2.2%

      \[\leadsto y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 45.5%

          \[\leadsto y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 92.6%

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

   11. Simplified 92.6%

       \[\leadsto y \cdot \color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+107}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+105}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 15: 91.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+107}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot {x}^{-0.5}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+105}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4e+107)
   (* (* y -0.3333333333333333) (pow x -0.5))
   (if (<= y 2.05e+105)
     (+ 1.0 (/ -0.1111111111111111 x))
     (* y (* -0.3333333333333333 (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4e+107) {
		tmp = (y * -0.3333333333333333) * pow(x, -0.5);
	} else if (y <= 2.05e+105) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y * (-0.3333333333333333 * pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4d+107)) then
        tmp = (y * (-0.3333333333333333d0)) * (x ** (-0.5d0))
    else if (y <= 2.05d+105) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = y * ((-0.3333333333333333d0) * (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4e+107) {
		tmp = (y * -0.3333333333333333) * Math.pow(x, -0.5);
	} else if (y <= 2.05e+105) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = y * (-0.3333333333333333 * Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4e+107:
		tmp = (y * -0.3333333333333333) * math.pow(x, -0.5)
	elif y <= 2.05e+105:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = y * (-0.3333333333333333 * math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4e+107)
		tmp = Float64(Float64(y * -0.3333333333333333) * (x ^ -0.5));
	elseif (y <= 2.05e+105)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(y * Float64(-0.3333333333333333 * (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4e+107)
		tmp = (y * -0.3333333333333333) * (x ^ -0.5);
	elseif (y <= 2.05e+105)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = y * (-0.3333333333333333 * (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4e+107], N[(N[(y * -0.3333333333333333), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e+105], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999999e107

   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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.5%

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

      5. metadata-eval 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 99.4%

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

   5. Taylor expanded in y around inf 96.0%

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

      1. associate-*r* 96.0%

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

   7. Simplified 96.0%

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

      1. expm1-log1p-u 94.2%

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

      2. expm1-udef 48.2%

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

      3. inv-pow 48.2%

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

      4. sqrt-pow1 48.2%

         \[\leadsto \left(-0.3333333333333333 \cdot y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 48.2%

         \[\leadsto \left(-0.3333333333333333 \cdot y\right) \cdot \left(e^{\mathsf{log1p}\left({x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   9. Applied egg-rr 48.2%

      \[\leadsto \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({x}^{-0.5}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 94.2%

          \[\leadsto \left(-0.3333333333333333 \cdot y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 96.0%

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

   11. Simplified 96.0%

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

   if -3.9999999999999999e107 < y < 2.0500000000000001e105

   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. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-/r* 99.8%

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

      8. metadata-eval 99.8%

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

      9. sub-neg 99.8%

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

      10. distribute-frac-neg 99.8%

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

      11. +-commutative 99.8%

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

      12. neg-mul-1 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-*r/ 99.8%

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

      15. fma-def 99.8%

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

      16. associate-/r* 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 92.2%

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

      1. cancel-sign-sub-inv 92.2%

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

      2. metadata-eval 92.2%

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

      3. associate-*r/ 92.2%

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

      4. metadata-eval 92.2%

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

   8. Simplified 92.2%

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

   if 2.0500000000000001e105 < 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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.5%

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

      5. metadata-eval 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.1%

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

      8. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around inf 96.5%

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

   5. Taylor expanded in y around inf 92.4%

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

      1. associate-*r* 92.4%

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

      2. *-commutative 92.4%

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

      3. associate-*l* 92.6%

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

   7. Simplified 92.6%

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

      1. expm1-log1p-u 45.5%

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

      2. expm1-udef 2.2%

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

      3. inv-pow 2.2%

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

      4. sqrt-pow1 2.2%

         \[\leadsto y \cdot \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \]

      5. metadata-eval 2.2%

         \[\leadsto y \cdot \left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{\color{blue}{-0.5}}\right)} - 1\right) \]

   9. Applied egg-rr 2.2%

      \[\leadsto y \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 45.5%

          \[\leadsto y \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\right)} \]

       2. expm1-log1p 92.6%

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

   11. Simplified 92.6%

       \[\leadsto y \cdot \color{blue}{\left(-0.3333333333333333 \cdot {x}^{-0.5}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+107}:\\ \;\;\;\;\left(y \cdot -0.3333333333333333\right) \cdot {x}^{-0.5}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+105}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.3333333333333333 \cdot {x}^{-0.5}\right)\\ \end{array} \]

Alternative 16: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+111} \lor \neg \left(y \leq 1.15 \cdot 10^{+104}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.1e+111) (not (<= y 1.15e+104)))
   (* y (/ -0.3333333333333333 (sqrt x)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.1e+111) || !(y <= 1.15e+104)) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / 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 <= (-4.1d+111)) .or. (.not. (y <= 1.15d+104))) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.1e+111) || !(y <= 1.15e+104)) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.1e+111) or not (y <= 1.15e+104):
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.1e+111) || !(y <= 1.15e+104))
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.1e+111) || ~((y <= 1.15e+104)))
		tmp = y * (-0.3333333333333333 / sqrt(x));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.1e+111], N[Not[LessEqual[y, 1.15e+104]], $MachinePrecision]], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.09999999999999986e111 or 1.14999999999999992e104 < 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. sub-neg 99.5%

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

      2. distribute-frac-neg 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.5%

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

      5. metadata-eval 99.5%

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

      6. neg-mul-1 99.5%

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

      7. times-frac 99.3%

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

      8. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around inf 97.0%

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

   5. Taylor expanded in y around inf 93.4%

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

      1. associate-*r* 93.4%

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

   7. Simplified 93.4%

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

      1. expm1-log1p-u 44.6%

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

      2. expm1-udef 44.6%

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

   9. Applied egg-rr 93.3%

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

       1. +-commutative 93.3%

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

       2. associate--l+ 93.3%

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

       3. metadata-eval 93.3%

          \[\leadsto -0.3333333333333333 \cdot \frac{y}{\sqrt{x}} + \color{blue}{0} \]

       4. +-rgt-identity 93.3%

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

       5. associate-*r/ 93.2%

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

       6. associate-/l* 93.3%

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

       7. associate-/r/ 93.4%

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

   11. Simplified 93.4%

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

   if -4.09999999999999986e111 < y < 1.14999999999999992e104

   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. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-/r* 99.8%

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

      8. metadata-eval 99.8%

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

      9. sub-neg 99.8%

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

      10. distribute-frac-neg 99.8%

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

      11. +-commutative 99.8%

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

      12. neg-mul-1 99.8%

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

      13. *-commutative 99.8%

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

      14. associate-*r/ 99.7%

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

      15. fma-def 99.7%

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

      16. associate-/r* 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 92.7%

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

      1. cancel-sign-sub-inv 92.7%

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

      2. metadata-eval 92.7%

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

      3. associate-*r/ 92.7%

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

      4. metadata-eval 92.7%

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

   8. Simplified 92.7%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+111} \lor \neg \left(y \leq 1.15 \cdot 10^{+104}\right):\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 17: 65.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\frac{0.012345679012345678}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.65e+137)
   (sqrt (/ 0.012345679012345678 (* x x)))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.65e+137) {
		tmp = sqrt((0.012345679012345678 / (x * x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / 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+137)) then
        tmp = sqrt((0.012345679012345678d0 / (x * x)))
    else
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.65e+137) {
		tmp = Math.sqrt((0.012345679012345678 / (x * x)));
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.65e+137:
		tmp = math.sqrt((0.012345679012345678 / (x * x)))
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.65e+137)
		tmp = sqrt(Float64(0.012345679012345678 / Float64(x * x)));
	else
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.65e+137)
		tmp = sqrt((0.012345679012345678 / (x * x)));
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.65e+137], N[Sqrt[N[(0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65000000000000001e137

   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. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-neg-frac 99.5%

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

      5. metadata-eval 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-/r* 99.5%

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

      8. metadata-eval 99.5%

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

      9. sub-neg 99.5%

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

      10. distribute-frac-neg 99.5%

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

      11. +-commutative 99.5%

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

      12. neg-mul-1 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-*r/ 99.3%

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

      15. fma-def 99.3%

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

      16. associate-/r* 99.3%

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

      17. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. fma-udef 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in x around 0 1.1%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 16.3%

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

      3. frac-times 16.2%

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

      4. metadata-eval 16.2%

         \[\leadsto \sqrt{\frac{\color{blue}{0.012345679012345678}}{x \cdot x}} \]

   8. Applied egg-rr 16.2%

      \[\leadsto \color{blue}{\sqrt{\frac{0.012345679012345678}{x \cdot x}}} \]

   if -1.65000000000000001e137 < y

   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. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-neg-frac 99.8%

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

      5. metadata-eval 99.8%

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

      6. *-commutative 99.8%

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

      7. associate-/r* 99.7%

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

      8. metadata-eval 99.7%

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

      9. sub-neg 99.7%

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

      10. distribute-frac-neg 99.7%

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

      11. +-commutative 99.7%

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

      12. neg-mul-1 99.7%

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

      13. *-commutative 99.7%

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

      14. associate-*r/ 99.7%

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

      15. fma-def 99.7%

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

      16. associate-/r* 99.7%

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

      17. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. fma-udef 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in y around 0 74.0%

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

      1. cancel-sign-sub-inv 74.0%

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

      2. metadata-eval 74.0%

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

      3. associate-*r/ 74.0%

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

      4. metadata-eval 74.0%

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

   8. Simplified 74.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+137}:\\ \;\;\;\;\sqrt{\frac{0.012345679012345678}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 18: 62.0% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.039:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.039) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.039) {
		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 <= 0.039d0) 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 <= 0.039) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.039:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.039)
		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 <= 0.039)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.039], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 0.0389999999999999999

   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. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.7%

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

      4. distribute-neg-frac 99.7%

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

      5. metadata-eval 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-/r* 99.6%

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

      8. metadata-eval 99.6%

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

      9. sub-neg 99.6%

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

      10. distribute-frac-neg 99.6%

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

      11. +-commutative 99.6%

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

      12. neg-mul-1 99.6%

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

      13. *-commutative 99.6%

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

      14. associate-*r/ 99.5%

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

      15. fma-def 99.5%

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

      16. associate-/r* 99.5%

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

      17. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. fma-udef 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in x around 0 64.6%

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

   if 0.0389999999999999999 < 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. sub-neg 99.8%

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

      2. distribute-frac-neg 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-/r* 99.8%

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

      5. metadata-eval 99.8%

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

      6. neg-mul-1 99.8%

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

      7. times-frac 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 98.5%

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

   5. Taylor expanded in y around 0 58.9%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.039:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 19: 63.2% 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 99.7%

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

   1. sub-neg 99.7%

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

   2. +-commutative 99.7%

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

   3. associate--l+ 99.7%

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

   4. distribute-neg-frac 99.7%

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

   5. metadata-eval 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-/r* 99.7%

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

   8. metadata-eval 99.7%

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

   9. sub-neg 99.7%

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

   10. distribute-frac-neg 99.7%

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

   11. +-commutative 99.7%

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

   12. neg-mul-1 99.7%

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

   13. *-commutative 99.7%

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

   14. associate-*r/ 99.6%

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

   15. fma-def 99.6%

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

   16. associate-/r* 99.6%

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

   17. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. fma-udef 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in y around 0 62.9%

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

   1. cancel-sign-sub-inv 62.9%

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

   2. metadata-eval 62.9%

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

   3. associate-*r/ 62.9%

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

   4. metadata-eval 62.9%

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

8. Simplified 62.9%

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

9. Final simplification 62.9%

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

Alternative 20: 31.8% 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 99.7%

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

   1. sub-neg 99.7%

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

   2. distribute-frac-neg 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-/r* 99.7%

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

   5. metadata-eval 99.7%

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

   6. neg-mul-1 99.7%

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

   7. times-frac 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around inf 65.7%

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

5. Taylor expanded in y around 0 29.7%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 29.7%

   \[\leadsto 1 \]

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 2023274 
(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)))))