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

Percentage Accurate: 99.7% → 99.7%

Time: 8.0s

Alternatives: 14

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

\[\begin{array}{l} \\ \left(1 + \frac{-1}{x \cdot 9}\right) - {x}^{-0.5} \cdot \frac{y}{3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (+ 1.0 (/ -1.0 (* x 9.0))) (* (pow x -0.5) (/ y 3.0))))

C

double code(double x, double y) {
	return (1.0 + (-1.0 / (x * 9.0))) - (pow(x, -0.5) * (y / 3.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 + ((-1.0d0) / (x * 9.0d0))) - ((x ** (-0.5d0)) * (y / 3.0d0))
end function

Java

public static double code(double x, double y) {
	return (1.0 + (-1.0 / (x * 9.0))) - (Math.pow(x, -0.5) * (y / 3.0));
}

Python

def code(x, y):
	return (1.0 + (-1.0 / (x * 9.0))) - (math.pow(x, -0.5) * (y / 3.0))

Julia

function code(x, y)
	return Float64(Float64(1.0 + Float64(-1.0 / Float64(x * 9.0))) - Float64((x ^ -0.5) * Float64(y / 3.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 + (-1.0 / (x * 9.0))) - ((x ^ -0.5) * (y / 3.0));
end

Wolfram

code[x_, y_] := N[(N[(1.0 + N[(-1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Power[x, -0.5], $MachinePrecision] * N[(y / 3.0), $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. *-un-lft-identity 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.7%

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

   4. pow1/2 99.7%

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

   5. pow-flip 99.8%

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

   6. metadata-eval 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 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 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(1 - \frac{0.1111111111111111}{x}\right) - {x}^{-0.5} \cdot \frac{y}{3} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (- (- 1.0 (/ 0.1111111111111111 x)) (* (pow x -0.5) (/ y 3.0))))

C

double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (pow(x, -0.5) * (y / 3.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - (0.1111111111111111d0 / x)) - ((x ** (-0.5d0)) * (y / 3.0d0))
end function

Java

public static double code(double x, double y) {
	return (1.0 - (0.1111111111111111 / x)) - (Math.pow(x, -0.5) * (y / 3.0));
}

Python

def code(x, y):
	return (1.0 - (0.1111111111111111 / x)) - (math.pow(x, -0.5) * (y / 3.0))

Julia

function code(x, y)
	return Float64(Float64(1.0 - Float64(0.1111111111111111 / x)) - Float64((x ^ -0.5) * Float64(y / 3.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - (0.1111111111111111 / x)) - ((x ^ -0.5) * (y / 3.0));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] - N[(N[Power[x, -0.5], $MachinePrecision] * N[(y / 3.0), $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. *-un-lft-identity 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.7%

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

   4. pow1/2 99.7%

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

   5. pow-flip 99.8%

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

   6. metadata-eval 99.8%

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

3. Applied egg-rr 99.8%

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

4. Taylor expanded in x around 0 99.7%

   \[\leadsto \left(1 - \color{blue}{\frac{0.1111111111111111}{x}}\right) - {x}^{-0.5} \cdot \frac{y}{3} \]

5. Final simplification 99.7%

   \[\leadsto \left(1 - \frac{0.1111111111111111}{x}\right) - {x}^{-0.5} \cdot \frac{y}{3} \]

Alternative 4: 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 5: 99.6% 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. 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. inv-pow 99.7%

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

   3. *-commutative 99.7%

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

   4. unpow-prod-down 99.6%

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

   5. metadata-eval 99.6%

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

   6. inv-pow 99.6%

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

   7. div-inv 99.7%

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

   8. *-commutative 99.7%

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

   9. metadata-eval 99.7%

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

   10. sqrt-prod 99.7%

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

3. Applied egg-rr 99.7%

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

   1. unsub-neg 99.7%

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

   2. sub-neg 99.7%

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

   3. distribute-neg-frac 99.7%

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

   4. metadata-eval 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 6: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+103} \lor \neg \left(y \leq 1.9 \cdot 10^{+32}\right):\\ \;\;\;\;y \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.2e+103) (not (<= y 1.9e+32)))
   (* y (* (pow x -0.5) -0.3333333333333333))
   (+ 1.0 (/ -0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -7.2e+103) || !(y <= 1.9e+32)) {
		tmp = y * (pow(x, -0.5) * -0.3333333333333333);
	} 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 <= (-7.2d+103)) .or. (.not. (y <= 1.9d+32))) then
        tmp = y * ((x ** (-0.5d0)) * (-0.3333333333333333d0))
    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 <= -7.2e+103) || !(y <= 1.9e+32)) {
		tmp = y * (Math.pow(x, -0.5) * -0.3333333333333333);
	} else {
		tmp = 1.0 + (-0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -7.2e+103) or not (y <= 1.9e+32):
		tmp = y * (math.pow(x, -0.5) * -0.3333333333333333)
	else:
		tmp = 1.0 + (-0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -7.2e+103) || !(y <= 1.9e+32))
		tmp = Float64(y * Float64((x ^ -0.5) * -0.3333333333333333));
	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 <= -7.2e+103) || ~((y <= 1.9e+32)))
		tmp = y * ((x ^ -0.5) * -0.3333333333333333);
	else
		tmp = 1.0 + (-0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -7.2e+103], N[Not[LessEqual[y, 1.9e+32]], $MachinePrecision]], N[(y * N[(N[Power[x, -0.5], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.20000000000000033e103 or 1.9000000000000002e32 < 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. *-un-lft-identity 99.6%

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

      2. *-commutative 99.6%

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

      3. times-frac 99.5%

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

      4. pow1/2 99.5%

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

      5. pow-flip 99.7%

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

      6. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in y around inf 90.7%

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

      1. associate-*r* 90.6%

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

      2. *-commutative 90.6%

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

      3. unpow1/2 90.6%

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

      4. unpow-1 90.6%

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

      5. exp-to-pow 86.6%

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

      6. *-commutative 86.6%

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

      7. log-pow 86.6%

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

      8. metadata-eval 86.6%

         \[\leadsto y \cdot \left(-0.3333333333333333 \cdot {\left(e^{\log \left({x}^{\color{blue}{\left(\frac{-2}{2}\right)}}\right)}\right)}^{0.5}\right) \]

      9. exp-prod 86.6%

         \[\leadsto y \cdot \left(-0.3333333333333333 \cdot \color{blue}{e^{\log \left({x}^{\left(\frac{-2}{2}\right)}\right) \cdot 0.5}}\right) \]

      10. metadata-eval 86.6%

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

      11. unpow-1 86.6%

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

      12. log-rec 86.6%

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

      13. distribute-lft-neg-out 86.6%

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

      14. distribute-rgt-neg-in 86.6%

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

      15. metadata-eval 86.6%

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

      16. exp-to-pow 90.6%

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

   8. Simplified 90.6%

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

   if -7.20000000000000033e103 < y < 1.9000000000000002e32

   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* 99.7%

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

      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 99.7%

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

      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 96.7%

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

      1. cancel-sign-sub-inv 96.7%

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

      2. metadata-eval 96.7%

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

      3. associate-*r/ 96.8%

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

      4. metadata-eval 96.8%

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

   6. Simplified 96.8%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+103} \lor \neg \left(y \leq 1.9 \cdot 10^{+32}\right):\\ \;\;\;\;y \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \end{array} \]

Alternative 7: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.12e+104)
   (* y (* (pow x -0.5) -0.3333333333333333))
   (if (<= y 1.9e+32)
     (+ 1.0 (/ -0.1111111111111111 x))
     (* -0.3333333333333333 (* (pow x -0.5) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.12e+104) {
		tmp = y * (pow(x, -0.5) * -0.3333333333333333);
	} else if (y <= 1.9e+32) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.3333333333333333 * (pow(x, -0.5) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.12d+104)) then
        tmp = y * ((x ** (-0.5d0)) * (-0.3333333333333333d0))
    else if (y <= 1.9d+32) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (-0.3333333333333333d0) * ((x ** (-0.5d0)) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.12e+104) {
		tmp = y * (Math.pow(x, -0.5) * -0.3333333333333333);
	} else if (y <= 1.9e+32) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.3333333333333333 * (Math.pow(x, -0.5) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.12e+104:
		tmp = y * (math.pow(x, -0.5) * -0.3333333333333333)
	elif y <= 1.9e+32:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = -0.3333333333333333 * (math.pow(x, -0.5) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.12e+104)
		tmp = Float64(y * Float64((x ^ -0.5) * -0.3333333333333333));
	elseif (y <= 1.9e+32)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(-0.3333333333333333 * Float64((x ^ -0.5) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.12e+104)
		tmp = y * ((x ^ -0.5) * -0.3333333333333333);
	elseif (y <= 1.9e+32)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = -0.3333333333333333 * ((x ^ -0.5) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.12e+104], N[(y * N[(N[Power[x, -0.5], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+32], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[x, -0.5], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.12000000000000003e104

   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. *-un-lft-identity 99.6%

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

      2. *-commutative 99.6%

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

      3. times-frac 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-flip 99.7%

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

      6. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in y around inf 97.4%

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

      1. associate-*r* 97.5%

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

      2. *-commutative 97.5%

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

      3. unpow1/2 97.5%

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

      4. unpow-1 97.5%

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

      5. exp-to-pow 92.6%

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

      6. *-commutative 92.6%

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

      7. log-pow 92.6%

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

      8. metadata-eval 92.6%

         \[\leadsto y \cdot \left(-0.3333333333333333 \cdot {\left(e^{\log \left({x}^{\color{blue}{\left(\frac{-2}{2}\right)}}\right)}\right)}^{0.5}\right) \]

      9. exp-prod 92.5%

         \[\leadsto y \cdot \left(-0.3333333333333333 \cdot \color{blue}{e^{\log \left({x}^{\left(\frac{-2}{2}\right)}\right) \cdot 0.5}}\right) \]

      10. metadata-eval 92.5%

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

      11. unpow-1 92.5%

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

      12. log-rec 92.5%

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

      13. distribute-lft-neg-out 92.5%

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

      14. distribute-rgt-neg-in 92.5%

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

      15. metadata-eval 92.5%

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

      16. exp-to-pow 97.5%

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

   8. Simplified 97.5%

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

   if -1.12000000000000003e104 < y < 1.9000000000000002e32

   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* 99.7%

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

      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 99.7%

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

      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 96.7%

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

      1. cancel-sign-sub-inv 96.7%

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

      2. metadata-eval 96.7%

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

      3. associate-*r/ 96.8%

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

      4. metadata-eval 96.8%

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

   6. Simplified 96.8%

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

   if 1.9000000000000002e32 < 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. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.4%

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

      4. pow1/2 99.4%

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

      5. pow-flip 99.7%

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

      6. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around inf 85.7%

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

      1. *-commutative 85.7%

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

   6. Simplified 85.7%

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

      1. pow1/2 85.7%

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

      2. inv-pow 85.7%

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

      3. pow-pow 85.7%

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

      4. metadata-eval 85.7%

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

      5. expm1-log1p-u 85.0%

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

      6. expm1-udef 43.3%

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

   8. Applied egg-rr 43.3%

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

      1. expm1-def 85.0%

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

      2. expm1-log1p 85.7%

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

   10. Simplified 85.7%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \left({x}^{-0.5} \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\\ \end{array} \]

Alternative 8: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+103}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.2e+103)
   (* (pow x -0.5) (* y -0.3333333333333333))
   (if (<= y 1.9e+32)
     (+ 1.0 (/ -0.1111111111111111 x))
     (* -0.3333333333333333 (* (pow x -0.5) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+103) {
		tmp = pow(x, -0.5) * (y * -0.3333333333333333);
	} else if (y <= 1.9e+32) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.3333333333333333 * (pow(x, -0.5) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.2d+103)) then
        tmp = (x ** (-0.5d0)) * (y * (-0.3333333333333333d0))
    else if (y <= 1.9d+32) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (-0.3333333333333333d0) * ((x ** (-0.5d0)) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+103) {
		tmp = Math.pow(x, -0.5) * (y * -0.3333333333333333);
	} else if (y <= 1.9e+32) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = -0.3333333333333333 * (Math.pow(x, -0.5) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2e+103:
		tmp = math.pow(x, -0.5) * (y * -0.3333333333333333)
	elif y <= 1.9e+32:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = -0.3333333333333333 * (math.pow(x, -0.5) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2e+103)
		tmp = Float64((x ^ -0.5) * Float64(y * -0.3333333333333333));
	elseif (y <= 1.9e+32)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(-0.3333333333333333 * Float64((x ^ -0.5) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.2e+103)
		tmp = (x ^ -0.5) * (y * -0.3333333333333333);
	elseif (y <= 1.9e+32)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = -0.3333333333333333 * ((x ^ -0.5) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2e+103], N[(N[Power[x, -0.5], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+32], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[x, -0.5], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.20000000000000033e103

   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. *-un-lft-identity 99.6%

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

      2. *-commutative 99.6%

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

      3. times-frac 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-flip 99.7%

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

      6. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around inf 97.4%

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

      1. *-commutative 97.4%

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

   6. Simplified 97.4%

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

      1. pow1/2 97.4%

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

      2. inv-pow 97.4%

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

      3. pow-pow 97.4%

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

      4. metadata-eval 97.4%

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

      5. expm1-log1p-u 94.5%

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

      6. expm1-udef 68.6%

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

   8. Applied egg-rr 68.6%

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

      1. expm1-def 94.5%

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

      2. expm1-log1p 97.4%

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

   10. Simplified 97.4%

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

       1. expm1-log1p-u 91.1%

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

       2. expm1-udef 91.1%

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

       3. associate-*l* 91.1%

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

   12. Applied egg-rr 91.1%

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

       1. expm1-def 91.1%

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

       2. expm1-log1p 97.5%

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

   14. Simplified 97.5%

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

   if -7.20000000000000033e103 < y < 1.9000000000000002e32

   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* 99.7%

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

      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 99.7%

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

      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 96.7%

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

      1. cancel-sign-sub-inv 96.7%

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

      2. metadata-eval 96.7%

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

      3. associate-*r/ 96.8%

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

      4. metadata-eval 96.8%

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

   6. Simplified 96.8%

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

   if 1.9000000000000002e32 < 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. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.4%

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

      4. pow1/2 99.4%

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

      5. pow-flip 99.7%

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

      6. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around inf 85.7%

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

      1. *-commutative 85.7%

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

   6. Simplified 85.7%

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

      1. pow1/2 85.7%

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

      2. inv-pow 85.7%

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

      3. pow-pow 85.7%

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

      4. metadata-eval 85.7%

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

      5. expm1-log1p-u 85.0%

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

      6. expm1-udef 43.3%

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

   8. Applied egg-rr 43.3%

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

      1. expm1-def 85.0%

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

      2. expm1-log1p 85.7%

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

   10. Simplified 85.7%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+103}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\\ \end{array} \]

Alternative 9: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+103}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.2e+103)
   (* (pow x -0.5) (* y -0.3333333333333333))
   (if (<= y 1.9e+32)
     (- 1.0 (pow (* x 9.0) -1.0))
     (* -0.3333333333333333 (* (pow x -0.5) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+103) {
		tmp = pow(x, -0.5) * (y * -0.3333333333333333);
	} else if (y <= 1.9e+32) {
		tmp = 1.0 - pow((x * 9.0), -1.0);
	} else {
		tmp = -0.3333333333333333 * (pow(x, -0.5) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.2d+103)) then
        tmp = (x ** (-0.5d0)) * (y * (-0.3333333333333333d0))
    else if (y <= 1.9d+32) then
        tmp = 1.0d0 - ((x * 9.0d0) ** (-1.0d0))
    else
        tmp = (-0.3333333333333333d0) * ((x ** (-0.5d0)) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.2e+103) {
		tmp = Math.pow(x, -0.5) * (y * -0.3333333333333333);
	} else if (y <= 1.9e+32) {
		tmp = 1.0 - Math.pow((x * 9.0), -1.0);
	} else {
		tmp = -0.3333333333333333 * (Math.pow(x, -0.5) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.2e+103:
		tmp = math.pow(x, -0.5) * (y * -0.3333333333333333)
	elif y <= 1.9e+32:
		tmp = 1.0 - math.pow((x * 9.0), -1.0)
	else:
		tmp = -0.3333333333333333 * (math.pow(x, -0.5) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.2e+103)
		tmp = Float64((x ^ -0.5) * Float64(y * -0.3333333333333333));
	elseif (y <= 1.9e+32)
		tmp = Float64(1.0 - (Float64(x * 9.0) ^ -1.0));
	else
		tmp = Float64(-0.3333333333333333 * Float64((x ^ -0.5) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.2e+103)
		tmp = (x ^ -0.5) * (y * -0.3333333333333333);
	elseif (y <= 1.9e+32)
		tmp = 1.0 - ((x * 9.0) ^ -1.0);
	else
		tmp = -0.3333333333333333 * ((x ^ -0.5) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.2e+103], N[(N[Power[x, -0.5], $MachinePrecision] * N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+32], N[(1.0 - N[Power[N[(x * 9.0), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], N[(-0.3333333333333333 * N[(N[Power[x, -0.5], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.20000000000000033e103

   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. *-un-lft-identity 99.6%

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

      2. *-commutative 99.6%

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

      3. times-frac 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-flip 99.7%

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

      6. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around inf 97.4%

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

      1. *-commutative 97.4%

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

   6. Simplified 97.4%

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

      1. pow1/2 97.4%

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

      2. inv-pow 97.4%

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

      3. pow-pow 97.4%

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

      4. metadata-eval 97.4%

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

      5. expm1-log1p-u 94.5%

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

      6. expm1-udef 68.6%

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

   8. Applied egg-rr 68.6%

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

      1. expm1-def 94.5%

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

      2. expm1-log1p 97.4%

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

   10. Simplified 97.4%

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

       1. expm1-log1p-u 91.1%

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

       2. expm1-udef 91.1%

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

       3. associate-*l* 91.1%

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

   12. Applied egg-rr 91.1%

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

       1. expm1-def 91.1%

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

       2. expm1-log1p 97.5%

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

   14. Simplified 97.5%

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

   if -7.20000000000000033e103 < y < 1.9000000000000002e32

   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. *-un-lft-identity 99.8%

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

      2. *-commutative 99.8%

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

      3. times-frac 99.8%

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

      4. pow1/2 99.8%

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

      5. pow-flip 99.8%

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

      6. metadata-eval 99.8%

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

   3. Applied egg-rr 99.8%

      \[\leadsto \left(1 - \frac{1}{x \cdot 9}\right) - \color{blue}{{x}^{-0.5} \cdot \frac{y}{3}} \]
   4. Step-by-step derivation

      1. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 96.7%

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

      1. associate-*r/ 96.8%

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

      2. metadata-eval 96.8%

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

   8. Simplified 96.8%

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

      1. div-inv 96.7%

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

      2. metadata-eval 96.7%

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

      3. inv-pow 96.7%

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

      4. unpow-prod-down 96.9%

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

      5. *-commutative 96.9%

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

   10. Applied egg-rr 96.9%

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

   if 1.9000000000000002e32 < 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. *-un-lft-identity 99.5%

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

      2. *-commutative 99.5%

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

      3. times-frac 99.4%

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

      4. pow1/2 99.4%

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

      5. pow-flip 99.7%

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

      6. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in y around inf 85.7%

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

      1. *-commutative 85.7%

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

   6. Simplified 85.7%

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

      1. pow1/2 85.7%

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

      2. inv-pow 85.7%

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

      3. pow-pow 85.7%

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

      4. metadata-eval 85.7%

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

      5. expm1-log1p-u 85.0%

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

      6. expm1-udef 43.3%

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

   8. Applied egg-rr 43.3%

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

      1. expm1-def 85.0%

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

      2. expm1-log1p 85.7%

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

   10. Simplified 85.7%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+103}:\\ \;\;\;\;{x}^{-0.5} \cdot \left(y \cdot -0.3333333333333333\right)\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;1 - {\left(x \cdot 9\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left({x}^{-0.5} \cdot y\right)\\ \end{array} \]

Alternative 10: 66.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \frac{0.012345679012345678}{x \cdot x}\\ \mathbf{if}\;y \leq -1.58 \cdot 10^{+112}:\\ \;\;\;\;\frac{t_0}{1 + -0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+132}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{1 + \frac{0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ 0.012345679012345678 (* x x)))))
   (if (<= y -1.58e+112)
     (/ t_0 (+ 1.0 (* -0.1111111111111111 (/ 1.0 x))))
     (if (<= y 2.1e+132)
       (+ 1.0 (/ -0.1111111111111111 x))
       (/ t_0 (+ 1.0 (/ 0.1111111111111111 x)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - (0.012345679012345678 / (x * x));
	double tmp;
	if (y <= -1.58e+112) {
		tmp = t_0 / (1.0 + (-0.1111111111111111 * (1.0 / x)));
	} else if (y <= 2.1e+132) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = t_0 / (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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (0.012345679012345678d0 / (x * x))
    if (y <= (-1.58d+112)) then
        tmp = t_0 / (1.0d0 + ((-0.1111111111111111d0) * (1.0d0 / x)))
    else if (y <= 2.1d+132) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = t_0 / (1.0d0 + (0.1111111111111111d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - (0.012345679012345678 / (x * x));
	double tmp;
	if (y <= -1.58e+112) {
		tmp = t_0 / (1.0 + (-0.1111111111111111 * (1.0 / x)));
	} else if (y <= 2.1e+132) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = t_0 / (1.0 + (0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - (0.012345679012345678 / (x * x))
	tmp = 0
	if y <= -1.58e+112:
		tmp = t_0 / (1.0 + (-0.1111111111111111 * (1.0 / x)))
	elif y <= 2.1e+132:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = t_0 / (1.0 + (0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - Float64(0.012345679012345678 / Float64(x * x)))
	tmp = 0.0
	if (y <= -1.58e+112)
		tmp = Float64(t_0 / Float64(1.0 + Float64(-0.1111111111111111 * Float64(1.0 / x))));
	elseif (y <= 2.1e+132)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(t_0 / Float64(1.0 + Float64(0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - (0.012345679012345678 / (x * x));
	tmp = 0.0;
	if (y <= -1.58e+112)
		tmp = t_0 / (1.0 + (-0.1111111111111111 * (1.0 / x)));
	elseif (y <= 2.1e+132)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = t_0 / (1.0 + (0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[(0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.58e+112], N[(t$95$0 / N[(1.0 + N[(-0.1111111111111111 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e+132], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(1.0 + N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.57999999999999992e112

   1. Initial program 99.6%

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

   3. Applied egg-rr 58.4%

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

   4. Taylor expanded in y around 0 2.7%

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

      1. associate-*r/ 2.7%

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

      2. metadata-eval 2.7%

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

      3. unpow2 2.7%

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

      4. associate-*r/ 2.7%

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

      5. metadata-eval 2.7%

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

   6. Simplified 2.7%

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

      1. add-sqr-sqrt 2.7%

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

      2. sqrt-unprod 2.7%

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

      3. frac-times 2.7%

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

      4. metadata-eval 2.7%

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

      5. metadata-eval 2.7%

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

      6. frac-times 2.7%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 23.5%

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

      9. div-inv 23.5%

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

   8. Applied egg-rr 23.5%

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

   if -1.57999999999999992e112 < y < 2.09999999999999993e132

   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* 99.7%

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

      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 99.7%

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

      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 89.6%

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

      1. cancel-sign-sub-inv 89.6%

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

      2. metadata-eval 89.6%

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

      3. associate-*r/ 89.7%

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

      4. metadata-eval 89.7%

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

   6. Simplified 89.7%

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

   if 2.09999999999999993e132 < y

   1. Initial program 99.5%

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

   3. Applied egg-rr 90.7%

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

   4. Taylor expanded in y around 0 17.9%

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

      1. associate-*r/ 17.9%

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

      2. metadata-eval 17.9%

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

      3. unpow2 17.9%

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

      4. associate-*r/ 17.9%

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

      5. metadata-eval 17.9%

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

   6. Simplified 17.9%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{0.1111111111111111}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.58 \cdot 10^{+112}:\\ \;\;\;\;\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + -0.1111111111111111 \cdot \frac{1}{x}}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{+132}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{0.1111111111111111}{x}}\\ \end{array} \]

Alternative 11: 64.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+132}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{0.1111111111111111}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e+132)
   (+ 1.0 (/ -0.1111111111111111 x))
   (/
    (- 1.0 (/ 0.012345679012345678 (* x x)))
    (+ 1.0 (/ 0.1111111111111111 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e+132) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (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 <= 2.1d+132) then
        tmp = 1.0d0 + ((-0.1111111111111111d0) / x)
    else
        tmp = (1.0d0 - (0.012345679012345678d0 / (x * x))) / (1.0d0 + (0.1111111111111111d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e+132) {
		tmp = 1.0 + (-0.1111111111111111 / x);
	} else {
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (1.0 + (0.1111111111111111 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e+132:
		tmp = 1.0 + (-0.1111111111111111 / x)
	else:
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (1.0 + (0.1111111111111111 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e+132)
		tmp = Float64(1.0 + Float64(-0.1111111111111111 / x));
	else
		tmp = Float64(Float64(1.0 - Float64(0.012345679012345678 / Float64(x * x))) / Float64(1.0 + Float64(0.1111111111111111 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e+132)
		tmp = 1.0 + (-0.1111111111111111 / x);
	else
		tmp = (1.0 - (0.012345679012345678 / (x * x))) / (1.0 + (0.1111111111111111 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e+132], N[(1.0 + N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(0.012345679012345678 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.09999999999999993e132

   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* 99.7%

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

      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 y around 0 71.8%

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

      1. cancel-sign-sub-inv 71.8%

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

      2. metadata-eval 71.8%

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

      3. associate-*r/ 71.9%

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

      4. metadata-eval 71.9%

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

   6. Simplified 71.9%

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

   if 2.09999999999999993e132 < y

   1. Initial program 99.5%

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

   3. Applied egg-rr 90.7%

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

   4. Taylor expanded in y around 0 17.9%

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

      1. associate-*r/ 17.9%

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

      2. metadata-eval 17.9%

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

      3. unpow2 17.9%

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

      4. associate-*r/ 17.9%

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

      5. metadata-eval 17.9%

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

   6. Simplified 17.9%

      \[\leadsto \color{blue}{\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{0.1111111111111111}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+132}:\\ \;\;\;\;1 + \frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{0.012345679012345678}{x \cdot x}}{1 + \frac{0.1111111111111111}{x}}\\ \end{array} \]

Alternative 12: 59.1% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e+26) (/ -0.1111111111111111 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e+26) {
		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 <= 5d+26) 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 <= 5e+26) {
		tmp = -0.1111111111111111 / x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5e+26:
		tmp = -0.1111111111111111 / x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e+26)
		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 <= 5e+26)
		tmp = -0.1111111111111111 / x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e+26], N[(-0.1111111111111111 / x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e26

   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 57.3%

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

   if 5.0000000000000001e26 < 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 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 64.9%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\frac{-0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 13: 61.8% 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. 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 y around 0 61.0%

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

   1. cancel-sign-sub-inv 61.0%

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

   2. metadata-eval 61.0%

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

   3. associate-*r/ 61.1%

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

   4. metadata-eval 61.1%

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

6. Simplified 61.1%

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

7. Final simplification 61.1%

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

Alternative 14: 31.6% 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 29.8%

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

5. Final simplification 29.8%

   \[\leadsto 1 \]

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 2023297 
(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)))))