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

Percentage Accurate: 99.7% → 99.7%

Time: 1.4s

Alternatives: 6

Speedup: 1.0×

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

Alternative 2: 92.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot 9}\\ t_1 := t\_0 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+92}:\\ \;\;\;\;1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* x 9.0))) (t_1 (- t_0 (/ y (* 3.0 (sqrt x))))))
   (if (<= y -6.8e+74) t_1 (if (<= y 6.5e+92) (- 1.0 t_0) t_1))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (x * 9.0);
	double t_1 = t_0 - (y / (3.0 * sqrt(x)));
	double tmp;
	if (y <= -6.8e+74) {
		tmp = t_1;
	} else if (y <= 6.5e+92) {
		tmp = 1.0 - t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (x * 9.0d0)
    t_1 = t_0 - (y / (3.0d0 * sqrt(x)))
    if (y <= (-6.8d+74)) then
        tmp = t_1
    else if (y <= 6.5d+92) then
        tmp = 1.0d0 - t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (x * 9.0);
	double t_1 = t_0 - (y / (3.0 * Math.sqrt(x)));
	double tmp;
	if (y <= -6.8e+74) {
		tmp = t_1;
	} else if (y <= 6.5e+92) {
		tmp = 1.0 - t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (x * 9.0)
	t_1 = t_0 - (y / (3.0 * math.sqrt(x)))
	tmp = 0
	if y <= -6.8e+74:
		tmp = t_1
	elif y <= 6.5e+92:
		tmp = 1.0 - t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(x * 9.0))
	t_1 = Float64(t_0 - Float64(y / Float64(3.0 * sqrt(x))))
	tmp = 0.0
	if (y <= -6.8e+74)
		tmp = t_1;
	elseif (y <= 6.5e+92)
		tmp = Float64(1.0 - t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (x * 9.0);
	t_1 = t_0 - (y / (3.0 * sqrt(x)));
	tmp = 0.0;
	if (y <= -6.8e+74)
		tmp = t_1;
	elseif (y <= 6.5e+92)
		tmp = 1.0 - t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.8e+74], t$95$1, If[LessEqual[y, 6.5e+92], N[(1.0 - t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.7999999999999998e74 or 6.49999999999999999e92 < y

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 91.6%

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

   if -6.7999999999999998e74 < y < 6.49999999999999999e92

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.1%

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

Alternative 3: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 9 - \frac{y}{3 \cdot \sqrt{x}}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{+118}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+98}:\\ \;\;\;\;1 - \frac{1}{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (* x 9.0) (/ y (* 3.0 (sqrt x))))))
   (if (<= y -7.5e+118)
     t_0
     (if (<= y 1.2e+98) (- 1.0 (/ 1.0 (* x 9.0))) t_0))))

C

double code(double x, double y) {
	double t_0 = (x * 9.0) - (y / (3.0 * sqrt(x)));
	double tmp;
	if (y <= -7.5e+118) {
		tmp = t_0;
	} else if (y <= 1.2e+98) {
		tmp = 1.0 - (1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	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 = (x * 9.0d0) - (y / (3.0d0 * sqrt(x)))
    if (y <= (-7.5d+118)) then
        tmp = t_0
    else if (y <= 1.2d+98) then
        tmp = 1.0d0 - (1.0d0 / (x * 9.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x * 9.0) - (y / (3.0 * Math.sqrt(x)));
	double tmp;
	if (y <= -7.5e+118) {
		tmp = t_0;
	} else if (y <= 1.2e+98) {
		tmp = 1.0 - (1.0 / (x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x * 9.0) - (y / (3.0 * math.sqrt(x)))
	tmp = 0
	if y <= -7.5e+118:
		tmp = t_0
	elif y <= 1.2e+98:
		tmp = 1.0 - (1.0 / (x * 9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x * 9.0) - Float64(y / Float64(3.0 * sqrt(x))))
	tmp = 0.0
	if (y <= -7.5e+118)
		tmp = t_0;
	elseif (y <= 1.2e+98)
		tmp = Float64(1.0 - Float64(1.0 / Float64(x * 9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x * 9.0) - (y / (3.0 * sqrt(x)));
	tmp = 0.0;
	if (y <= -7.5e+118)
		tmp = t_0;
	elseif (y <= 1.2e+98)
		tmp = 1.0 - (1.0 / (x * 9.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x * 9.0), $MachinePrecision] - N[(y / N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e+118], t$95$0, If[LessEqual[y, 1.2e+98], N[(1.0 - N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.50000000000000003e118 or 1.1999999999999999e98 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 71.6%

      \[\leadsto \color{blue}{x \cdot 9} - \frac{y}{3 \cdot \sqrt{x}} \]

   if -7.50000000000000003e118 < y < 1.1999999999999999e98

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.1%

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

Alternative 4: 62.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 62.2%

   \[\leadsto \color{blue}{1 - \frac{1}{x \cdot 9}} \]
6. Add Preprocessing

Alternative 5: 3.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt x))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return Math.sqrt(x);
}

Python

def code(x, y):
	return math.sqrt(x)

Julia

function code(x, y)
	return sqrt(x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Sqrt[x], $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 62.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 3.0%

   \[\leadsto x \cdot \color{blue}{9} \]

9. Taylor expanded in x around inf

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

11. Applied rewrites 3.5%

    \[\leadsto \color{blue}{\sqrt{x}} \]
12. Add Preprocessing

Alternative 6: 3.2% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ x \cdot 9 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 9.0))

C

double code(double x, double y) {
	return x * 9.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return x * 9.0;
}

Python

def code(x, y):
	return x * 9.0

Julia

function code(x, y)
	return Float64(x * 9.0)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 9.0;
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 62.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 3.0%

   \[\leadsto x \cdot \color{blue}{9} \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.9× 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 2024321 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, D"
  :precision binary64
  :pre (TRUE)

  :alt
  (! :herbie-platform default (- (- 1 (/ (/ 1 x) 9)) (/ y (* 3 (sqrt x)))))

  (- (- 1.0 (/ 1.0 (* x 9.0))) (/ y (* 3.0 (sqrt x)))))