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

Percentage Accurate: 99.7% → 99.7%

Time: 10.6s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. add-exp-log_binary64 46.3%

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

3. Applied rewrite-once 46.3%

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

   1. rem-exp-log 99.7%

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

   2. associate--l- 99.7%

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

   3. associate-/r* 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. associate-/r* 99.7%

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

   2. clear-num 99.6%

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

   3. associate-/r/ 99.6%

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

   4. pow1/2 99.6%

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

   5. pow-flip 99.7%

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

   6. metadata-eval 99.7%

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

   7. div-inv 99.6%

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

   8. metadata-eval 99.6%

      \[\leadsto 1 - \left(\frac{0.1111111111111111}{x} + {x}^{-0.5} \cdot \left(y \cdot \color{blue}{0.3333333333333333}\right)\right) \]

5. Applied egg-rr 99.6%

   \[\leadsto 1 - \left(\frac{0.1111111111111111}{x} + \color{blue}{{x}^{-0.5} \cdot \left(y \cdot 0.3333333333333333\right)}\right) \]
6. Step-by-step derivation

   1. metadata-eval 99.6%

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

   2. div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) * 0.3333333333333333d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 - N[(N[(0.1111111111111111 / x), $MachinePrecision] + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * 0.3333333333333333), $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. associate--l- 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/r* 99.7%

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

   3. div-inv 99.6%

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

   4. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate--l- 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(1.0 - N[(N[(0.1111111111111111 / x), $MachinePrecision] + N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / 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. associate--l- 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-/r* 99.7%

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

   3. div-inv 99.6%

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

   4. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

   1. metadata-eval 99.6%

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

   2. div-inv 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 6: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{+100} \lor \neg \left(y \leq 1.9 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{{x}^{-0.5}}{\frac{-3}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.72e+100) (not (<= y 1.9e+32)))
   (/ (pow x -0.5) (/ -3.0 y))
   (- 1.0 (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.72e+100) || !(y <= 1.9e+32)) {
		tmp = pow(x, -0.5) / (-3.0 / y);
	} else {
		tmp = 1.0 - (0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.72d+100)) .or. (.not. (y <= 1.9d+32))) then
        tmp = (x ** (-0.5d0)) / ((-3.0d0) / y)
    else
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.72e+100) || !(y <= 1.9e+32)) {
		tmp = Math.pow(x, -0.5) / (-3.0 / y);
	} else {
		tmp = 1.0 - (0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.72e+100) or not (y <= 1.9e+32):
		tmp = math.pow(x, -0.5) / (-3.0 / y)
	else:
		tmp = 1.0 - (0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.72e+100) || !(y <= 1.9e+32))
		tmp = Float64((x ^ -0.5) / Float64(-3.0 / y));
	else
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.72e+100) || ~((y <= 1.9e+32)))
		tmp = (x ^ -0.5) / (-3.0 / y);
	else
		tmp = 1.0 - (0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.72e+100], N[Not[LessEqual[y, 1.9e+32]], $MachinePrecision]], N[(N[Power[x, -0.5], $MachinePrecision] / N[(-3.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7200000000000001e100 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. add-exp-log_binary64 44.6%

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

   3. Applied rewrite-once 44.6%

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

      1. rem-exp-log 99.6%

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

      2. associate--l- 99.6%

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

      3. associate-/r* 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 90.0%

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

      1. associate-*r* 89.9%

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

      2. *-commutative 89.9%

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

      3. *-commutative 89.9%

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

   8. Simplified 89.9%

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

      1. *-commutative 89.9%

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

      2. sqrt-div 89.8%

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

      3. metadata-eval 89.8%

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

      4. un-div-inv 89.9%

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

   10. Applied egg-rr 89.9%

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

       1. associate-*r/ 89.9%

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

       2. associate-*l/ 90.0%

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

       3. metadata-eval 90.0%

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

       4. metadata-eval 90.0%

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

       5. div-inv 90.2%

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

       6. div-inv 90.1%

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

       7. metadata-eval 90.1%

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

       8. sqrt-div 90.1%

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

       9. *-commutative 90.1%

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

       10. associate-/l* 90.1%

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

       11. remove-double-neg 90.1%

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

       12. frac-2neg 90.1%

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

       13. inv-pow 90.1%

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

       14. metadata-eval 90.1%

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

       15. sqrt-pow1 90.2%

           \[\leadsto \frac{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}{\frac{3}{-y}} \]

       16. metadata-eval 90.2%

           \[\leadsto \frac{{x}^{\left(\frac{\color{blue}{-1}}{2}\right)}}{\frac{3}{-y}} \]

       17. metadata-eval 90.2%

           \[\leadsto \frac{{x}^{\color{blue}{-0.5}}}{\frac{3}{-y}} \]

       18. frac-2neg 90.2%

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

       19. metadata-eval 90.2%

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

       20. remove-double-neg 90.2%

           \[\leadsto \frac{{x}^{-0.5}}{\frac{-3}{\color{blue}{y}}} \]

   12. Applied egg-rr 90.2%

       \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{\frac{-3}{y}}} \]

   if -1.7200000000000001e100 < 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. associate--l- 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 97.4%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{+100} \lor \neg \left(y \leq 1.9 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{{x}^{-0.5}}{\frac{-3}{y}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \]

Alternative 7: 91.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -7.2e+103) (not (<= y 1.9e+32)))
   (* y (/ -0.3333333333333333 (sqrt x)))
   (- 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 * (-0.3333333333333333 / sqrt(x));
	} else {
		tmp = 1.0 - (0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-7.2d+103)) .or. (.not. (y <= 1.9d+32))) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -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. add-exp-log_binary64 43.6%

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

   3. Applied rewrite-once 43.6%

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

      1. rem-exp-log 99.6%

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

      2. associate--l- 99.6%

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

      3. associate-/r* 99.6%

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

   5. Simplified 99.6%

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

   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. *-commutative 90.6%

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

   8. Simplified 90.6%

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

      1. *-commutative 90.6%

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

      2. sqrt-div 90.6%

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

      3. metadata-eval 90.6%

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

      4. un-div-inv 90.6%

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

   10. Applied egg-rr 90.6%

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

   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. associate--l- 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 96.8%

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

4. Final simplification 94.3%

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

Alternative 8: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{+100} \lor \neg \left(y \leq 1.9 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.72e+100) (not (<= y 1.9e+32)))
   (/ (/ y (sqrt x)) -3.0)
   (- 1.0 (/ 0.1111111111111111 x))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.72e+100) || !(y <= 1.9e+32)) {
		tmp = (y / sqrt(x)) / -3.0;
	} else {
		tmp = 1.0 - (0.1111111111111111 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.72d+100)) .or. (.not. (y <= 1.9d+32))) then
        tmp = (y / sqrt(x)) / (-3.0d0)
    else
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.72e+100) || !(y <= 1.9e+32)) {
		tmp = (y / Math.sqrt(x)) / -3.0;
	} else {
		tmp = 1.0 - (0.1111111111111111 / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.72e+100) or not (y <= 1.9e+32):
		tmp = (y / math.sqrt(x)) / -3.0
	else:
		tmp = 1.0 - (0.1111111111111111 / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.72e+100) || !(y <= 1.9e+32))
		tmp = Float64(Float64(y / sqrt(x)) / -3.0);
	else
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.72e+100) || ~((y <= 1.9e+32)))
		tmp = (y / sqrt(x)) / -3.0;
	else
		tmp = 1.0 - (0.1111111111111111 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.72e+100], N[Not[LessEqual[y, 1.9e+32]], $MachinePrecision]], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7200000000000001e100 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. add-exp-log_binary64 44.6%

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

   3. Applied rewrite-once 44.6%

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

      1. rem-exp-log 99.6%

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

      2. associate--l- 99.6%

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

      3. associate-/r* 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in y around inf 90.0%

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

      1. associate-*r* 89.9%

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

      2. *-commutative 89.9%

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

      3. *-commutative 89.9%

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

   8. Simplified 89.9%

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

      1. *-commutative 89.9%

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

      2. sqrt-div 89.8%

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

      3. metadata-eval 89.8%

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

      4. un-div-inv 89.9%

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

   10. Applied egg-rr 89.9%

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

       1. associate-*r/ 89.9%

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

       2. associate-*l/ 90.0%

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

       3. metadata-eval 90.0%

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

       4. metadata-eval 90.0%

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

       5. div-inv 90.2%

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

       6. metadata-eval 90.2%

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

   12. Applied egg-rr 90.2%

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

   if -1.7200000000000001e100 < 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. associate--l- 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 97.4%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{+100} \lor \neg \left(y \leq 1.9 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{\frac{y}{\sqrt{x}}}{-3}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \end{array} \]

Alternative 9: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+109}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.7e+109)
   (* y (/ -0.3333333333333333 (sqrt x)))
   (if (<= y 1.9e+32)
     (- 1.0 (/ 0.1111111111111111 x))
     (* (/ y (sqrt x)) -0.3333333333333333))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.7e+109) {
		tmp = y * (-0.3333333333333333 / sqrt(x));
	} else if (y <= 1.9e+32) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = (y / sqrt(x)) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.7d+109)) then
        tmp = y * ((-0.3333333333333333d0) / sqrt(x))
    else if (y <= 1.9d+32) then
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    else
        tmp = (y / sqrt(x)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.7e+109) {
		tmp = y * (-0.3333333333333333 / Math.sqrt(x));
	} else if (y <= 1.9e+32) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = (y / Math.sqrt(x)) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.7e+109:
		tmp = y * (-0.3333333333333333 / math.sqrt(x))
	elif y <= 1.9e+32:
		tmp = 1.0 - (0.1111111111111111 / x)
	else:
		tmp = (y / math.sqrt(x)) * -0.3333333333333333
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.7e+109)
		tmp = Float64(y * Float64(-0.3333333333333333 / sqrt(x)));
	elseif (y <= 1.9e+32)
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	else
		tmp = Float64(Float64(y / sqrt(x)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.7e+109)
		tmp = y * (-0.3333333333333333 / sqrt(x));
	elseif (y <= 1.9e+32)
		tmp = 1.0 - (0.1111111111111111 / x);
	else
		tmp = (y / sqrt(x)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.7e+109], N[(y * N[(-0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+32], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.70000000000000001e109

   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. add-exp-log_binary64 93.1%

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

   3. Applied rewrite-once 93.1%

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

      1. rem-exp-log 99.6%

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

      2. associate--l- 99.6%

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

      3. associate-/r* 99.8%

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

   5. Simplified 99.8%

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

   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. *-commutative 97.5%

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

   8. Simplified 97.5%

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

      1. *-commutative 97.5%

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

      2. sqrt-div 97.6%

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

      3. metadata-eval 97.6%

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

      4. un-div-inv 97.5%

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

   10. Applied egg-rr 97.5%

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

   if -2.70000000000000001e109 < 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. associate--l- 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 96.8%

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

   if 1.9000000000000002e32 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf 85.7%

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

      1. *-commutative 85.7%

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

   4. Simplified 85.7%

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

      1. *-commutative 85.7%

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

      2. sqrt-div 85.7%

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

      3. metadata-eval 85.7%

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

      4. div-inv 85.7%

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

      5. remove-double-neg 85.7%

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

      6. neg-sub0 85.7%

         \[\leadsto \color{blue}{\left(0 - \left(-\frac{y}{\sqrt{x}}\right)\right)} \cdot -0.3333333333333333 \]

      7. frac-2neg 85.7%

         \[\leadsto \left(0 - \left(-\color{blue}{\frac{-y}{-\sqrt{x}}}\right)\right) \cdot -0.3333333333333333 \]

      8. distribute-frac-neg 85.7%

         \[\leadsto \left(0 - \left(-\color{blue}{\left(-\frac{y}{-\sqrt{x}}\right)}\right)\right) \cdot -0.3333333333333333 \]

      9. remove-double-neg 85.7%

         \[\leadsto \left(0 - \color{blue}{\frac{y}{-\sqrt{x}}}\right) \cdot -0.3333333333333333 \]

   6. Applied egg-rr 85.7%

      \[\leadsto \color{blue}{\left(0 - \frac{y}{-\sqrt{x}}\right)} \cdot -0.3333333333333333 \]
   7. Step-by-step derivation

      1. sub0-neg 85.7%

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

      2. distribute-frac-neg 85.7%

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

      3. neg-mul-1 85.7%

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

      4. neg-mul-1 85.7%

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

      5. times-frac 85.7%

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

      6. metadata-eval 85.7%

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

      7. *-lft-identity 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+109}:\\ \;\;\;\;y \cdot \frac{-0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 10: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{+100}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.72e+100)
   (/ y (* (sqrt x) -3.0))
   (if (<= y 1.9e+32)
     (- 1.0 (/ 0.1111111111111111 x))
     (* (/ y (sqrt x)) -0.3333333333333333))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.72e+100) {
		tmp = y / (sqrt(x) * -3.0);
	} else if (y <= 1.9e+32) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = (y / sqrt(x)) * -0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.72d+100)) then
        tmp = y / (sqrt(x) * (-3.0d0))
    else if (y <= 1.9d+32) then
        tmp = 1.0d0 - (0.1111111111111111d0 / x)
    else
        tmp = (y / sqrt(x)) * (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.72e+100) {
		tmp = y / (Math.sqrt(x) * -3.0);
	} else if (y <= 1.9e+32) {
		tmp = 1.0 - (0.1111111111111111 / x);
	} else {
		tmp = (y / Math.sqrt(x)) * -0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.72e+100:
		tmp = y / (math.sqrt(x) * -3.0)
	elif y <= 1.9e+32:
		tmp = 1.0 - (0.1111111111111111 / x)
	else:
		tmp = (y / math.sqrt(x)) * -0.3333333333333333
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.72e+100)
		tmp = Float64(y / Float64(sqrt(x) * -3.0));
	elseif (y <= 1.9e+32)
		tmp = Float64(1.0 - Float64(0.1111111111111111 / x));
	else
		tmp = Float64(Float64(y / sqrt(x)) * -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.72e+100)
		tmp = y / (sqrt(x) * -3.0);
	elseif (y <= 1.9e+32)
		tmp = 1.0 - (0.1111111111111111 / x);
	else
		tmp = (y / sqrt(x)) * -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.72e+100], N[(y / N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e+32], N[(1.0 - N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7200000000000001e100

   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. add-exp-log_binary64 93.2%

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

   3. Applied rewrite-once 93.2%

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

      1. rem-exp-log 99.6%

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

      2. associate--l- 99.6%

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

      3. associate-/r* 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around inf 95.5%

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

      1. associate-*r* 95.6%

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

      2. *-commutative 95.6%

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

      3. *-commutative 95.6%

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

   8. Simplified 95.6%

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

      1. *-commutative 95.6%

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

      2. sqrt-div 95.7%

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

      3. metadata-eval 95.7%

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

      4. un-div-inv 95.6%

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

   10. Applied egg-rr 95.6%

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

       1. associate-*r/ 95.5%

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

       2. associate-*l/ 95.5%

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

       3. metadata-eval 95.5%

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

       4. metadata-eval 95.5%

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

       5. distribute-rgt-neg-in 95.5%

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

       6. metadata-eval 95.5%

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

       7. metadata-eval 95.5%

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

       8. div-inv 95.8%

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

       9. neg-sub0 95.8%

          \[\leadsto \color{blue}{0 - \frac{\frac{y}{\sqrt{x}}}{3}} \]

       10. associate-/l/ 95.8%

           \[\leadsto 0 - \color{blue}{\frac{y}{3 \cdot \sqrt{x}}} \]

       11. div-inv 95.6%

           \[\leadsto 0 - \color{blue}{y \cdot \frac{1}{3 \cdot \sqrt{x}}} \]

       12. associate-/r* 95.6%

           \[\leadsto 0 - y \cdot \color{blue}{\frac{\frac{1}{3}}{\sqrt{x}}} \]

       13. metadata-eval 95.6%

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

   12. Applied egg-rr 95.6%

       \[\leadsto \color{blue}{0 - y \cdot \frac{0.3333333333333333}{\sqrt{x}}} \]
   13. Step-by-step derivation

       1. sub0-neg 95.6%

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

       2. neg-mul-1 95.6%

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

       3. associate-*r/ 95.5%

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

       4. metadata-eval 95.5%

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

       5. times-frac 95.5%

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

       6. *-commutative 95.5%

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

       7. *-rgt-identity 95.5%

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

       8. associate-/r* 95.5%

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

       9. associate-/l* 95.9%

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

       10. metadata-eval 95.9%

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

       11. associate-/l/ 95.8%

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

   14. Simplified 95.8%

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

   if -1.7200000000000001e100 < 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. associate--l- 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/r* 99.7%

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

      4. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 97.4%

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

   if 1.9000000000000002e32 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf 85.7%

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

      1. *-commutative 85.7%

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

   4. Simplified 85.7%

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

      1. *-commutative 85.7%

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

      2. sqrt-div 85.7%

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

      3. metadata-eval 85.7%

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

      4. div-inv 85.7%

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

      5. remove-double-neg 85.7%

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

      6. neg-sub0 85.7%

         \[\leadsto \color{blue}{\left(0 - \left(-\frac{y}{\sqrt{x}}\right)\right)} \cdot -0.3333333333333333 \]

      7. frac-2neg 85.7%

         \[\leadsto \left(0 - \left(-\color{blue}{\frac{-y}{-\sqrt{x}}}\right)\right) \cdot -0.3333333333333333 \]

      8. distribute-frac-neg 85.7%

         \[\leadsto \left(0 - \left(-\color{blue}{\left(-\frac{y}{-\sqrt{x}}\right)}\right)\right) \cdot -0.3333333333333333 \]

      9. remove-double-neg 85.7%

         \[\leadsto \left(0 - \color{blue}{\frac{y}{-\sqrt{x}}}\right) \cdot -0.3333333333333333 \]

   6. Applied egg-rr 85.7%

      \[\leadsto \color{blue}{\left(0 - \frac{y}{-\sqrt{x}}\right)} \cdot -0.3333333333333333 \]
   7. Step-by-step derivation

      1. sub0-neg 85.7%

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

      2. distribute-frac-neg 85.7%

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

      3. neg-mul-1 85.7%

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

      4. neg-mul-1 85.7%

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

      5. times-frac 85.7%

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

      6. metadata-eval 85.7%

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

      7. *-lft-identity 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{+100}:\\ \;\;\;\;\frac{y}{\sqrt{x} \cdot -3}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+32}:\\ \;\;\;\;1 - \frac{0.1111111111111111}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\sqrt{x}} \cdot -0.3333333333333333\\ \end{array} \]

Alternative 11: 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. 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. 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 12: 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. associate--l- 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-/r* 99.7%

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

   4. metadata-eval 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in x around 0 61.1%

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

5. Final simplification 61.1%

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

Alternative 13: 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. Taylor expanded in x around inf 29.8%

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

3. 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)))))