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

Percentage Accurate: 100.0% → 100.0%

Time: 2.0s

Alternatives: 6

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 0.04481 - x\\ \mathbf{if}\;x \leq -2.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* x 0.04481) x)))
   (if (<= x -2.05)
     t_0
     (if (<= x 1.15)
       (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
       t_0))))

C

double code(double x) {
	double t_0 = (x * 0.04481) - x;
	double tmp;
	if (x <= -2.05) {
		tmp = t_0;
	} else if (x <= 1.15) {
		tmp = (2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * 0.04481d0) - x
    if (x <= (-2.05d0)) then
        tmp = t_0
    else if (x <= 1.15d0) then
        tmp = (2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * 0.04481) - x;
	double tmp;
	if (x <= -2.05) {
		tmp = t_0;
	} else if (x <= 1.15) {
		tmp = (2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * 0.04481) - x
	tmp = 0
	if x <= -2.05:
		tmp = t_0
	elif x <= 1.15:
		tmp = (2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * 0.04481) - x)
	tmp = 0.0
	if (x <= -2.05)
		tmp = t_0;
	elseif (x <= 1.15)
		tmp = Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * 0.04481) - x;
	tmp = 0.0;
	if (x <= -2.05)
		tmp = t_0;
	elseif (x <= 1.15)
		tmp = (2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * 0.04481), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, -2.05], t$95$0, If[LessEqual[x, 1.15], N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0499999999999998 or 1.1499999999999999 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{230753}{100000} + \frac{-20191289437}{10000000000} \cdot x\right)} - x \]

   5. Applied rewrites 24.8%

      \[\leadsto \color{blue}{x \cdot 0.04481} - x \]

   if -2.0499999999999998 < x < 1.1499999999999999

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x} \]

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 60.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 0.04481 - x\\ \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;2.30753 + x \cdot 0.27061\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (* x 0.04481) x)))
   (if (<= x -1.05) t_0 (if (<= x 1.15) (+ 2.30753 (* x 0.27061)) t_0))))

C

double code(double x) {
	double t_0 = (x * 0.04481) - x;
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 1.15) {
		tmp = 2.30753 + (x * 0.27061);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x * 0.04481d0) - x
    if (x <= (-1.05d0)) then
        tmp = t_0
    else if (x <= 1.15d0) then
        tmp = 2.30753d0 + (x * 0.27061d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x * 0.04481) - x;
	double tmp;
	if (x <= -1.05) {
		tmp = t_0;
	} else if (x <= 1.15) {
		tmp = 2.30753 + (x * 0.27061);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x * 0.04481) - x
	tmp = 0
	if x <= -1.05:
		tmp = t_0
	elif x <= 1.15:
		tmp = 2.30753 + (x * 0.27061)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x * 0.04481) - x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 1.15)
		tmp = Float64(2.30753 + Float64(x * 0.27061));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x * 0.04481) - x;
	tmp = 0.0;
	if (x <= -1.05)
		tmp = t_0;
	elseif (x <= 1.15)
		tmp = 2.30753 + (x * 0.27061);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * 0.04481), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, -1.05], t$95$0, If[LessEqual[x, 1.15], N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{230753}{100000} + \frac{-20191289437}{10000000000} \cdot x\right)} - x \]

   5. Applied rewrites 24.8%

      \[\leadsto \color{blue}{x \cdot 0.04481} - x \]

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right)} \]

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{2.30753 + x \cdot 0.27061} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 48.7% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ 2.30753 + x \cdot 0.27061 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 2.30753 (* x 0.27061)))

C

double code(double x) {
	return 2.30753 + (x * 0.27061);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.30753d0 + (x * 0.27061d0)
end function

Java

public static double code(double x) {
	return 2.30753 + (x * 0.27061);
}

Python

def code(x):
	return 2.30753 + (x * 0.27061)

Julia

function code(x)
	return Float64(2.30753 + Float64(x * 0.27061))
end

MATLAB

function tmp = code(x)
	tmp = 2.30753 + (x * 0.27061);
end

Wolfram

code[x_] := N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right)} \]

5. Applied rewrites 50.8%

   \[\leadsto \color{blue}{2.30753 + x \cdot 0.27061} \]
6. Add Preprocessing

Alternative 5: 2.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.27061 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.27061))

C

double code(double x) {
	return x * 0.27061;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.27061d0
end function

Java

public static double code(double x) {
	return x * 0.27061;
}

Python

def code(x):
	return x * 0.27061

Julia

function code(x)
	return Float64(x * 0.27061)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.27061;
end

Wolfram

code[x_] := N[(x * 0.27061), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)} \]

5. Applied rewrites 2.2%

   \[\leadsto \color{blue}{x \cdot 0.27061} \]
6. Add Preprocessing

Alternative 6: 2.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.04481 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.04481))

C

double code(double x) {
	return x * 0.04481;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.04481d0
end function

Java

public static double code(double x) {
	return x * 0.04481;
}

Python

def code(x):
	return x * 0.04481

Julia

function code(x)
	return Float64(x * 0.04481)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.04481;
end

Wolfram

code[x_] := N[(x * 0.04481), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x} \]

5. Applied rewrites 51.3%

   \[\leadsto \color{blue}{\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{230753}{100000} + \color{blue}{x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)} \]

8. Applied rewrites 2.2%

   \[\leadsto x \cdot \color{blue}{0.04481} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  :pre (TRUE)
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))