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

Percentage Accurate: 100.0% → 100.0%

Time: 4.7s

Alternatives: 6

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x - \frac{2.30753 + x \cdot 0.27061}{1 + \left(0.99229 + x \cdot 0.04481\right) \cdot x} \]
2. Step-by-step derivation

   1. div-inv 100.0%

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

   2. +-commutative 100.0%

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

   3. fma-def 100.0%

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

   4. +-commutative 100.0%

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

   5. *-commutative 100.0%

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

   6. fma-def 100.0%

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

   7. +-commutative 100.0%

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

   8. fma-def 100.0%

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

3. Applied egg-rr 100.0%

   \[\leadsto x - \color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right) \cdot \frac{1}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)}} \]
4. Step-by-step derivation

   1. fma-udef 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 3: 98.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 98.5%

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

   1. *-commutative 98.5%

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

4. Simplified 98.5%

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

5. Final simplification 98.5%

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

Alternative 4: 98.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x -3.6) x (if (<= x 1.2) -2.30753 x)))

C

double code(double x) {
	double tmp;
	if (x <= -3.6) {
		tmp = x;
	} else if (x <= 1.2) {
		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 <= 1.2d0) 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 <= 1.2) {
		tmp = -2.30753;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -3.6:
		tmp = x
	elif x <= 1.2:
		tmp = -2.30753
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -3.6)
		tmp = x;
	elseif (x <= 1.2)
		tmp = -2.30753;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -3.6)
		tmp = x;
	elseif (x <= 1.2)
		tmp = -2.30753;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -3.6], x, If[LessEqual[x, 1.2], -2.30753, x]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000009 or 1.19999999999999996 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.3%

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

   3. Taylor expanded in x around inf 98.6%

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

   if -3.60000000000000009 < x < 1.19999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.2%

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

   3. Taylor expanded in x around 0 97.1%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;-2.30753\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 97.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x - 2.30753

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x - 2.30753), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0 97.2%

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

3. Final simplification 97.2%

   \[\leadsto x - 2.30753 \]

Alternative 6: 50.7% 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%

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

2. Taylor expanded in x around 0 97.2%

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

3. Taylor expanded in x around 0 50.8%

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

4. Final simplification 50.8%

   \[\leadsto -2.30753 \]

Reproduce

herbie shell --seed 2023181 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, D"
  :precision binary64
  (- x (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* (+ 0.99229 (* x 0.04481)) x)))))