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

Percentage Accurate: 99.9% → 99.9%

Time: 4.1s

Alternatives: 5

Speedup: 1.1×

• 24.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot x - 3}{6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (* x x) 3.0) 6.0))

C

double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) - 3.0d0) / 6.0d0
end function

Java

public static double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

Python

def code(x):
	return ((x * x) - 3.0) / 6.0

Julia

function code(x)
	return Float64(Float64(Float64(x * x) - 3.0) / 6.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) - 3.0) / 6.0;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] - 3.0), $MachinePrecision] / 6.0), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot x - 3}{6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (* x x) 3.0) 6.0))

C

double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) - 3.0d0) / 6.0d0
end function

Java

public static double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}

Python

def code(x):
	return ((x * x) - 3.0) / 6.0

Julia

function code(x)
	return Float64(Float64(Float64(x * x) - 3.0) / 6.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) - 3.0) / 6.0;
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] - 3.0), $MachinePrecision] / 6.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, x, -3\right)}{6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fma x x -3.0) 6.0))

C

double code(double x) {
	return fma(x, x, -3.0) / 6.0;
}

Julia

function code(x)
	return Float64(fma(x, x, -3.0) / 6.0)
end

Wolfram

code[x_] := N[(N[(x * x + -3.0), $MachinePrecision] / 6.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\frac{x \cdot x - 3}{6} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{{x}^{2} - 3}}{6} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{{x}^{2} - \color{blue}{-1 \cdot -3}}{6} \]

   2. metadata-eval N/A

      \[\leadsto \frac{{x}^{2} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot -3}{6} \]

   3. metadata-eval N/A

      \[\leadsto \frac{{x}^{2} - \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}}{6} \]

   4. rgt-mult-inverse N/A

      \[\leadsto \frac{{x}^{2} - \left(\mathsf{neg}\left(\color{blue}{{x}^{2} \cdot \frac{1}{{x}^{2}}}\right)\right) \cdot \left(\mathsf{neg}\left(3\right)\right)}{6} \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \frac{\color{blue}{{x}^{2} + \left({x}^{2} \cdot \frac{1}{{x}^{2}}\right) \cdot \left(\mathsf{neg}\left(3\right)\right)}}{6} \]

   6. associate-*r* N/A

      \[\leadsto \frac{{x}^{2} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}}{6} \]

   7. *-commutative N/A

      \[\leadsto \frac{{x}^{2} + {x}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{1}{{x}^{2}}\right)}}{6} \]

   8. unpow2 N/A

      \[\leadsto \frac{\color{blue}{x \cdot x} + {x}^{2} \cdot \left(\left(\mathsf{neg}\left(3\right)\right) \cdot \frac{1}{{x}^{2}}\right)}{6} \]

   9. *-commutative N/A

      \[\leadsto \frac{x \cdot x + {x}^{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(3\right)\right)\right)}}{6} \]

   10. associate-*r* N/A

       \[\leadsto \frac{x \cdot x + \color{blue}{\left({x}^{2} \cdot \frac{1}{{x}^{2}}\right) \cdot \left(\mathsf{neg}\left(3\right)\right)}}{6} \]

   11. rgt-mult-inverse N/A

       \[\leadsto \frac{x \cdot x + \color{blue}{1} \cdot \left(\mathsf{neg}\left(3\right)\right)}{6} \]

   12. metadata-eval N/A

       \[\leadsto \frac{x \cdot x + 1 \cdot \color{blue}{-3}}{6} \]

   13. metadata-eval N/A

       \[\leadsto \frac{x \cdot x + \color{blue}{-3}}{6} \]

   14. metadata-eval N/A

       \[\leadsto \frac{x \cdot x + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}}{6} \]

   15. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, \mathsf{neg}\left(3\right)\right)}}{6} \]

   16. metadata-eval 99.9

       \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{-3}\right)}{6} \]

5. Applied rewrites 99.9%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -3\right)}}{6} \]
6. Add Preprocessing

Alternative 2: 74.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7) -0.5 (* (* x x) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = -0.5;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.7d0) then
        tmp = -0.5d0
    else
        tmp = (x * x) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = -0.5;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = -0.5
	else:
		tmp = (x * x) * 0.16666666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = -0.5;
	else
		tmp = Float64(Float64(x * x) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = -0.5;
	else
		tmp = (x * x) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], -0.5, N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.9%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{-0.5} \]

   if 1.69999999999999996 < x

   1. Initial program 99.8%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{6}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{6}} \]

      3. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{6} \]

      4. lower-*.f64 98.0

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot 0.16666666666666666 \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.16666666666666666} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 74.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.16666666666666666\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7) -0.5 (* (* x 0.16666666666666666) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = -0.5;
	} else {
		tmp = (x * 0.16666666666666666) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.7d0) then
        tmp = -0.5d0
    else
        tmp = (x * 0.16666666666666666d0) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = -0.5;
	} else {
		tmp = (x * 0.16666666666666666) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = -0.5
	else:
		tmp = (x * 0.16666666666666666) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = -0.5;
	else
		tmp = Float64(Float64(x * 0.16666666666666666) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = -0.5;
	else
		tmp = (x * 0.16666666666666666) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], -0.5, N[(N[(x * 0.16666666666666666), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.9%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1}{2}} \]
   4. Step-by-step derivation

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{-0.5} \]

   if 1.69999999999999996 < x

   1. Initial program 99.8%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{6}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{1}{6}} \]

      3. unpow2 N/A

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{6} \]

      4. lower-*.f64 98.0

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot 0.16666666666666666 \]

   5. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.16666666666666666} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.7%

      \[\leadsto \sqrt{0.027777777777777776 \cdot {x}^{4}} \]
   8. Step-by-step derivation

   9. Applied rewrites 97.9%

      \[\leadsto \left(x \cdot 0.16666666666666666\right) \cdot \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.16666666666666666 \cdot x, x, -0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* 0.16666666666666666 x) x -0.5))

C

double code(double x) {
	return fma((0.16666666666666666 * x), x, -0.5);
}

Julia

function code(x)
	return fma(Float64(0.16666666666666666 * x), x, -0.5)
end

Wolfram

code[x_] := N[(N[(0.16666666666666666 * x), $MachinePrecision] * x + -0.5), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\frac{x \cdot x - 3}{6} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} - \frac{1}{2}} \]
4. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{1}{6} \cdot {x}^{2} - \color{blue}{-1 \cdot \frac{-1}{2}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{1}{6} \cdot {x}^{2} - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \cdot \frac{-1}{2} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1}{6} \cdot {x}^{2} - \left(\mathsf{neg}\left(1\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   4. rgt-mult-inverse N/A

      \[\leadsto \frac{1}{6} \cdot {x}^{2} - \left(\mathsf{neg}\left(\color{blue}{{x}^{2} \cdot \frac{1}{{x}^{2}}}\right)\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   5. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} + \left({x}^{2} \cdot \frac{1}{{x}^{2}}\right) \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{1}{6} \cdot {x}^{2} + \color{blue}{{x}^{2} \cdot \left(\frac{1}{{x}^{2}} \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{6} \cdot {x}^{2} + {x}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{{x}^{2}}\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{1}{6} \cdot {x}^{2} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{2}} \]

   9. associate-*l* N/A

      \[\leadsto \frac{1}{6} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(\frac{1}{{x}^{2}} \cdot {x}^{2}\right)} \]

   10. lft-mult-inverse N/A

       \[\leadsto \frac{1}{6} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \color{blue}{1} \]

   11. metadata-eval N/A

       \[\leadsto \frac{1}{6} \cdot {x}^{2} + \color{blue}{\frac{-1}{2}} \cdot 1 \]

   12. metadata-eval N/A

       \[\leadsto \frac{1}{6} \cdot {x}^{2} + \color{blue}{\frac{-1}{2}} \]

   13. metadata-eval N/A

       \[\leadsto \frac{1}{6} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(x \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   15. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \color{blue}{\left(x \cdot \frac{1}{6}\right)} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   17. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \frac{1}{6}, x, \mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

   18. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x}, x, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   19. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot x}, x, \mathsf{neg}\left(\frac{1}{2}\right)\right) \]

   20. metadata-eval 99.8

       \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot x, x, \color{blue}{-0.5}\right) \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666 \cdot x, x, -0.5\right)} \]
6. Add Preprocessing

Alternative 5: 50.2% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -0.5)

C

double code(double x) {
	return -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -0.5d0
end function

Java

public static double code(double x) {
	return -0.5;
}

Python

def code(x):
	return -0.5

Julia

function code(x)
	return -0.5
end

MATLAB

function tmp = code(x)
	tmp = -0.5;
end

Wolfram

code[x_] := -0.5

Derivation

1. Initial program 99.9%

   \[\frac{x \cdot x - 3}{6} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 47.6%

   \[\leadsto \color{blue}{-0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024338 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))