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

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 6

Speedup: 1.0×

Specification

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + \frac{x \cdot \left(0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)\right)}{0.99229 + x \cdot -0.04481}} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  (/
   (+ 2.30753 (* x 0.27061))
   (+
    1.0
    (/
     (* x (- 0.9846394441 (* 0.0020079361 (* x x))))
     (+ 0.99229 (* x -0.04481)))))
  x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + ((x * (0.9846394441 - (0.0020079361 * (x * 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 * (0.9846394441d0 - (0.0020079361d0 * (x * 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 * (0.9846394441 - (0.0020079361 * (x * x)))) / (0.99229 + (x * -0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + ((x * (0.9846394441 - (0.0020079361 * (x * 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(Float64(x * Float64(0.9846394441 - Float64(0.0020079361 * Float64(x * x)))) / Float64(0.99229 + Float64(x * -0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + ((x * (0.9846394441 - (0.0020079361 * (x * 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[(N[(x * N[(0.9846394441 - N[(0.0020079361 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. flip-+ 100.0%

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

   3. associate-*l/ 100.0%

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

   4. metadata-eval 100.0%

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(\color{blue}{0.9846394441} - \left(x \cdot 0.04481\right) \cdot \left(x \cdot 0.04481\right)\right) \cdot x}{0.99229 - x \cdot 0.04481}} - x \]

   5. *-commutative 100.0%

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - \color{blue}{\left(0.04481 \cdot x\right)} \cdot \left(x \cdot 0.04481\right)\right) \cdot x}{0.99229 - x \cdot 0.04481}} - x \]

   6. *-commutative 100.0%

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - \left(0.04481 \cdot x\right) \cdot \color{blue}{\left(0.04481 \cdot x\right)}\right) \cdot x}{0.99229 - x \cdot 0.04481}} - x \]

   7. swap-sqr 100.0%

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - \color{blue}{\left(0.04481 \cdot 0.04481\right) \cdot \left(x \cdot x\right)}\right) \cdot x}{0.99229 - x \cdot 0.04481}} - x \]

   8. metadata-eval 100.0%

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - \color{blue}{0.0020079361} \cdot \left(x \cdot x\right)\right) \cdot x}{0.99229 - x \cdot 0.04481}} - x \]

   9. *-commutative 100.0%

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)\right) \cdot x}{0.99229 - \color{blue}{0.04481 \cdot x}}} - x \]

   10. cancel-sign-sub-inv 100.0%

       \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)\right) \cdot x}{\color{blue}{0.99229 + \left(-0.04481\right) \cdot x}}} - x \]

   11. metadata-eval 100.0%

       \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \frac{\left(0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)\right) \cdot x}{0.99229 + \color{blue}{-0.04481} \cdot x}} - x \]

3. Applied egg-rr 100.0%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{\frac{\left(0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)\right) \cdot x}{0.99229 + -0.04481 \cdot x}}} - x \]

4. Final simplification 100.0%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \frac{x \cdot \left(0.9846394441 - 0.0020079361 \cdot \left(x \cdot x\right)\right)}{0.99229 + x \cdot -0.04481}} - x \]

Alternative 2: 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. Final simplification 100.0%

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

Alternative 3: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot 0.99229} - x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * 0.99229), $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. Taylor expanded in x around 0 98.1%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} - x \]
3. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} - x \]

4. Simplified 98.1%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} - x \]

5. Final simplification 98.1%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot 0.99229} - x \]

Alternative 4: 98.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -3.6) (- x) (if (<= x 1.15) 2.30753 (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -x;
	} else if (x <= 1.15) {
		tmp = 2.30753;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-3.6d0)) then
        tmp = -x
    else if (x <= 1.15d0) then
        tmp = 2.30753d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = -x;
	} else if (x <= 1.15) {
		tmp = 2.30753;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.6:
		tmp = -x
	elif x <= 1.15:
		tmp = 2.30753
	else:
		tmp = -x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.6)
		tmp = Float64(-x);
	elseif (x <= 1.15)
		tmp = 2.30753;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.6)
		tmp = -x;
	elseif (x <= 1.15)
		tmp = 2.30753;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.6], (-x), If[LessEqual[x, 1.15], 2.30753, (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009 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. Taylor expanded in x around 0 97.9%

      \[\leadsto \color{blue}{2.30753} - x \]

   3. Taylor expanded in x around inf 99.2%

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

      1. neg-mul-1 99.2%

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

   5. Simplified 99.2%

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

   if -3.60000000000000009 < 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. Taylor expanded in x around 0 97.7%

      \[\leadsto \color{blue}{\left(2.30753 + -2.0191289437 \cdot x\right)} - x \]

   3. Taylor expanded in x around 0 97.7%

      \[\leadsto \color{blue}{2.30753 + -3.0191289437 \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative 97.7%

         \[\leadsto 2.30753 + \color{blue}{x \cdot -3.0191289437} \]

   5. Simplified 97.7%

      \[\leadsto \color{blue}{2.30753 + x \cdot -3.0191289437} \]

   6. Taylor expanded in x around 0 96.5%

      \[\leadsto \color{blue}{2.30753} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;2.30753\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 5: 97.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ 2.30753 - x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 2.30753 x))

C

double code(double x) {
	return 2.30753 - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.30753d0 - x
end function

Java

public static double code(double x) {
	return 2.30753 - x;
}

Python

def code(x):
	return 2.30753 - x

Julia

function code(x)
	return Float64(2.30753 - x)
end

MATLAB

function tmp = code(x)
	tmp = 2.30753 - x;
end

Wolfram

code[x_] := N[(2.30753 - 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. Taylor expanded in x around 0 97.3%

   \[\leadsto \color{blue}{2.30753} - x \]

3. Final simplification 97.3%

   \[\leadsto 2.30753 - x \]

Alternative 6: 51.4% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ 2.30753 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.30753)

C

double code(double x) {
	return 2.30753;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.30753

Julia

function code(x)
	return 2.30753
end

MATLAB

function tmp = code(x)
	tmp = 2.30753;
end

Wolfram

code[x_] := 2.30753

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. Taylor expanded in x around 0 54.9%

   \[\leadsto \color{blue}{\left(2.30753 + -2.0191289437 \cdot x\right)} - x \]

3. Taylor expanded in x around 0 54.9%

   \[\leadsto \color{blue}{2.30753 + -3.0191289437 \cdot x} \]
4. Step-by-step derivation

   1. *-commutative 54.9%

      \[\leadsto 2.30753 + \color{blue}{x \cdot -3.0191289437} \]

5. Simplified 54.9%

   \[\leadsto \color{blue}{2.30753 + x \cdot -3.0191289437} \]

6. Taylor expanded in x around 0 46.7%

   \[\leadsto \color{blue}{2.30753} \]

7. Final simplification 46.7%

   \[\leadsto 2.30753 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))