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

Percentage Accurate: 99.9% → 99.9%

Time: 4.0s

Alternatives: 5

Speedup: 1.1×

• 24.6% 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

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{x}{6}, x, -0.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (/ x 6.0) x -0.5))

C

double code(double x) {
	return fma((x / 6.0), x, -0.5);
}

Julia

function code(x)
	return fma(Float64(x / 6.0), x, -0.5)
end

Wolfram

code[x_] := N[(N[(x / 6.0), $MachinePrecision] * x + -0.5), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\frac{x \cdot x - 3}{6} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. div-inv N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 0.16666666666666666, x, -0.5\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. lower-/.f64 99.9

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{6}}, x, -0.5\right) \]

6. Applied rewrites 99.9%

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

Alternative 2: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(0.16666666666666666 \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 1.0) -0.5 (* (* 0.16666666666666666 x) x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 1.0) {
		tmp = -0.5;
	} else {
		tmp = (0.16666666666666666 * x) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 1.0d0) then
        tmp = -0.5d0
    else
        tmp = (0.16666666666666666d0 * x) * x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 1.0) {
		tmp = -0.5;
	} else {
		tmp = (0.16666666666666666 * x) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 1.0:
		tmp = -0.5
	else:
		tmp = (0.16666666666666666 * x) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 1.0)
		tmp = -0.5;
	else
		tmp = Float64(Float64(0.16666666666666666 * x) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 1.0)
		tmp = -0.5;
	else
		tmp = (0.16666666666666666 * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.0], -0.5, N[(N[(0.16666666666666666 * x), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.4%

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

   if 1 < (*.f64 x x)

   1. Initial program 99.8%

      \[\frac{x \cdot x - 3}{6} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. sqr-pow N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 99.5

          \[\leadsto {\left({\left(\frac{6}{\mathsf{fma}\left(x, x, -3\right)}\right)}^{\color{blue}{-0.5}}\right)}^{2} \]

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{{\left({\left(\frac{6}{\mathsf{fma}\left(x, x, -3\right)}\right)}^{-0.5}\right)}^{2}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{{x}^{2} \cdot {\left(\sqrt{\frac{1}{6}}\right)}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. rem-square-sqrt N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 98.8

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

   7. Applied rewrites 98.8%

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

   9. Applied rewrites 98.9%

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

Alternative 3: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 3.0) -0.5 (* (* x x) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 3.0) {
		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 * x) <= 3.0d0) 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 * x) <= 3.0) {
		tmp = -0.5;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 3.0:
		tmp = -0.5
	else:
		tmp = (x * x) * 0.16666666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 3.0)
		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 * x) <= 3.0)
		tmp = -0.5;
	else
		tmp = (x * x) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 3.0], -0.5, N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.4%

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

   if 3 < (*.f64 x 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. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]
5. 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. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. sub-neg N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. div-inv N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 5: 50.8% 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 48.5%

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

Reproduce

herbie shell --seed 2024249 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))