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

Percentage Accurate: 99.7% → 99.7%

Time: 8.8s

Alternatives: 10

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: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+33} \lor \neg \left(y \leq 3.4 \cdot 10^{+37}\right):\\ \;\;\;\;1 - \sqrt{\frac{1}{x}} \cdot \left(y \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.12e+33) (not (<= y 3.4e+37)))
   (- 1.0 (* (sqrt (/ 1.0 x)) (* y 0.3333333333333333)))
   (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.12e+33) || !(y <= 3.4e+37)) {
		tmp = 1.0 - (sqrt((1.0 / x)) * (y * 0.3333333333333333));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.12d+33)) .or. (.not. (y <= 3.4d+37))) then
        tmp = 1.0d0 - (sqrt((1.0d0 / x)) * (y * 0.3333333333333333d0))
    else
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.12e+33) || !(y <= 3.4e+37)) {
		tmp = 1.0 - (Math.sqrt((1.0 / x)) * (y * 0.3333333333333333));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.12e+33) or not (y <= 3.4e+37):
		tmp = 1.0 - (math.sqrt((1.0 / x)) * (y * 0.3333333333333333))
	else:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.12e+33) || !(y <= 3.4e+37))
		tmp = Float64(1.0 - Float64(sqrt(Float64(1.0 / x)) * Float64(y * 0.3333333333333333)));
	else
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.12e+33) || ~((y <= 3.4e+37)))
		tmp = 1.0 - (sqrt((1.0 / x)) * (y * 0.3333333333333333));
	else
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.12e+33], N[Not[LessEqual[y, 3.4e+37]], $MachinePrecision]], N[(1.0 - N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(y * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.12e33 or 3.40000000000000006e37 < 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. *-commutative 99.6%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

      5. distribute-frac-neg 99.6%

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

      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. Step-by-step derivation

      1. associate-+l- 99.5%

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

      2. cancel-sign-sub-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. metadata-eval 99.5%

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

      5. times-frac 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. div-inv 99.5%

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

      8. associate-/r* 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. sqrt-div 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-def 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around inf 93.5%

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

      1. *-commutative 93.5%

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

      2. associate-*l* 93.5%

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

      3. *-commutative 93.5%

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

   10. Simplified 93.5%

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

   if -1.12e33 < y < 3.40000000000000006e37

   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. *-commutative 99.8%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      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 99.7%

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

      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 y around 0 98.8%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+33} \lor \neg \left(y \leq 3.4 \cdot 10^{+37}\right):\\ \;\;\;\;1 - \sqrt{\frac{1}{x}} \cdot \left(y \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]

Alternative 3: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+31}:\\ \;\;\;\;1 - y \cdot \frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+37}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.55e+31)
   (- 1.0 (* y (/ 0.3333333333333333 (sqrt x))))
   (if (<= y 2.2e+37)
     (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))
     (- 1.0 (* 0.3333333333333333 (* y (sqrt (/ 1.0 x))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.55e+31) {
		tmp = 1.0 - (y * (0.3333333333333333 / sqrt(x)));
	} else if (y <= 2.2e+37) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = 1.0 - (0.3333333333333333 * (y * sqrt((1.0 / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.55d+31)) then
        tmp = 1.0d0 - (y * (0.3333333333333333d0 / sqrt(x)))
    else if (y <= 2.2d+37) then
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    else
        tmp = 1.0d0 - (0.3333333333333333d0 * (y * sqrt((1.0d0 / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.55e+31) {
		tmp = 1.0 - (y * (0.3333333333333333 / Math.sqrt(x)));
	} else if (y <= 2.2e+37) {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	} else {
		tmp = 1.0 - (0.3333333333333333 * (y * Math.sqrt((1.0 / x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.55e+31:
		tmp = 1.0 - (y * (0.3333333333333333 / math.sqrt(x)))
	elif y <= 2.2e+37:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	else:
		tmp = 1.0 - (0.3333333333333333 * (y * math.sqrt((1.0 / x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.55e+31)
		tmp = Float64(1.0 - Float64(y * Float64(0.3333333333333333 / sqrt(x))));
	elseif (y <= 2.2e+37)
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	else
		tmp = Float64(1.0 - Float64(0.3333333333333333 * Float64(y * sqrt(Float64(1.0 / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.55e+31)
		tmp = 1.0 - (y * (0.3333333333333333 / sqrt(x)));
	elseif (y <= 2.2e+37)
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	else
		tmp = 1.0 - (0.3333333333333333 * (y * sqrt((1.0 / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.55e+31], N[(1.0 - N[(y * N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+37], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.3333333333333333 * N[(y * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5500000000000001e31

   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. *-commutative 99.7%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

      5. distribute-frac-neg 99.6%

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

      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. Step-by-step derivation

      1. associate-+l- 99.5%

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

      2. cancel-sign-sub-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. metadata-eval 99.5%

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

      5. times-frac 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. div-inv 99.5%

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

      8. associate-/r* 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. sqrt-div 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-def 99.6%

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

   7. Simplified 99.6%

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

      1. fma-udef 99.6%

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

      2. +-commutative 99.6%

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

      3. add-sqr-sqrt 99.5%

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

      4. distribute-rgt-out 99.6%

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

      5. sqrt-div 99.5%

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

      6. metadata-eval 99.5%

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

      7. sqrt-div 99.5%

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

      8. metadata-eval 99.5%

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

   9. Applied egg-rr 99.5%

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

   10. Taylor expanded in x around inf 94.9%

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

   if -1.5500000000000001e31 < y < 2.2000000000000001e37

   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. *-commutative 99.8%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      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 99.7%

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

      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 y around 0 98.8%

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

   if 2.2000000000000001e37 < 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. *-commutative 99.5%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      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 99.5%

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

      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. Step-by-step derivation

      1. associate-+l- 99.5%

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

      2. cancel-sign-sub-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. metadata-eval 99.5%

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

      5. times-frac 99.5%

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

      6. *-un-lft-identity 99.5%

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

      7. div-inv 99.5%

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

      8. associate-/r* 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. sqrt-div 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 91.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+31}:\\ \;\;\;\;1 - y \cdot \frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+37}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 - 0.3333333333333333 \cdot \left(y \cdot \sqrt{\frac{1}{x}}\right)\\ \end{array} \]

Alternative 4: 95.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+33} \lor \neg \left(y \leq 3.2 \cdot 10^{+37}\right):\\ \;\;\;\;1 - y \cdot \frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.6e+33) (not (<= y 3.2e+37)))
   (- 1.0 (* y (/ 0.3333333333333333 (sqrt x))))
   (+ 1.0 (* 0.1111111111111111 (/ -1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e+33) || !(y <= 3.2e+37)) {
		tmp = 1.0 - (y * (0.3333333333333333 / sqrt(x)));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.6d+33)) .or. (.not. (y <= 3.2d+37))) then
        tmp = 1.0d0 - (y * (0.3333333333333333d0 / sqrt(x)))
    else
        tmp = 1.0d0 + (0.1111111111111111d0 * ((-1.0d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.6e+33) || !(y <= 3.2e+37)) {
		tmp = 1.0 - (y * (0.3333333333333333 / Math.sqrt(x)));
	} else {
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.6e+33) or not (y <= 3.2e+37):
		tmp = 1.0 - (y * (0.3333333333333333 / math.sqrt(x)))
	else:
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.6e+33) || !(y <= 3.2e+37))
		tmp = Float64(1.0 - Float64(y * Float64(0.3333333333333333 / sqrt(x))));
	else
		tmp = Float64(1.0 + Float64(0.1111111111111111 * Float64(-1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.6e+33) || ~((y <= 3.2e+37)))
		tmp = 1.0 - (y * (0.3333333333333333 / sqrt(x)));
	else
		tmp = 1.0 + (0.1111111111111111 * (-1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.6e+33], N[Not[LessEqual[y, 3.2e+37]], $MachinePrecision]], N[(1.0 - N[(y * N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(0.1111111111111111 * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.60000000000000009e33 or 3.20000000000000014e37 < 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. *-commutative 99.6%

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

      3. associate-/r* 99.6%

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

      4. metadata-eval 99.6%

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

      5. distribute-frac-neg 99.6%

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

      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. Step-by-step derivation

      1. associate-+l- 99.5%

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

      2. cancel-sign-sub-inv 99.5%

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

      3. metadata-eval 99.5%

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

      4. metadata-eval 99.5%

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

      5. times-frac 99.6%

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

      6. *-un-lft-identity 99.6%

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

      7. div-inv 99.5%

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

      8. associate-/r* 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. sqrt-div 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-def 99.6%

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

   7. Simplified 99.6%

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

      1. fma-udef 99.6%

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

      2. +-commutative 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. distribute-rgt-out 99.6%

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

      5. sqrt-div 99.5%

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

      6. metadata-eval 99.5%

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

      7. sqrt-div 99.5%

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

      8. metadata-eval 99.5%

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

   9. Applied egg-rr 99.5%

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

   10. Taylor expanded in x around inf 93.5%

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

   if -1.60000000000000009e33 < y < 3.20000000000000014e37

   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. *-commutative 99.8%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

      5. distribute-frac-neg 99.7%

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

      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 99.7%

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

      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 y around 0 98.8%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+33} \lor \neg \left(y \leq 3.2 \cdot 10^{+37}\right):\\ \;\;\;\;1 - y \cdot \frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;1 + 0.1111111111111111 \cdot \frac{-1}{x}\\ \end{array} \]

Alternative 5: 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. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   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 6: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   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. Step-by-step derivation

   1. associate-+l- 99.6%

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

   2. cancel-sign-sub-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. metadata-eval 99.6%

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

   5. times-frac 99.7%

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

   6. *-un-lft-identity 99.7%

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

   7. div-inv 99.6%

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

   8. associate-/r* 99.6%

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

   9. metadata-eval 99.6%

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

   10. metadata-eval 99.6%

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

   11. sqrt-div 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 7: 62.9% accurate, 16.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(1 - \frac{1}{x \cdot 9}\right) - \frac{y}{3 \cdot \sqrt{x}} \]
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. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   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 y around 0 59.8%

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

5. Final simplification 59.8%

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

Alternative 8: 61.3% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1700000000000:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1700000000000.0) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1700000000000.0) {
		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 <= 1700000000000.0d0) 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 <= 1700000000000.0) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1700000000000.0:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1700000000000.0)
		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 <= 1700000000000.0)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1700000000000.0], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e12

   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. *-commutative 99.6%

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

      3. associate-/r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. distribute-frac-neg 99.5%

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

      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 99.5%

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

      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 54.5%

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

   if 1.7e12 < 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. *-commutative 99.8%

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

      3. associate-/r* 99.8%

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

      4. metadata-eval 99.8%

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

      5. distribute-frac-neg 99.8%

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

      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 64.1%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1700000000000:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 63.0% 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%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   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 y around 0 59.8%

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

   1. associate-*r/ 59.8%

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

   2. metadata-eval 59.8%

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

6. Simplified 59.8%

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

7. Final simplification 59.8%

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

Alternative 10: 32.2% 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. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

   5. distribute-frac-neg 99.7%

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

   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 32.9%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 32.9%

   \[\leadsto 1 \]

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 2023321 
(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)))))