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

Percentage Accurate: 99.8% → 99.9%

Time: 6.0s

Alternatives: 5

Speedup: 1.7×

• 24.1% 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: 99.8% 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 99.8

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot 0.16666666666666666, x, -0.5\right)} \]
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: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 2e-7) -0.5 (* 0.16666666666666666 (* x x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-7) {
		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) <= 2d-7) 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) <= 2e-7) {
		tmp = -0.5;
	} else {
		tmp = 0.16666666666666666 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 2e-7:
		tmp = -0.5
	else:
		tmp = 0.16666666666666666 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-7)
		tmp = -0.5;
	else
		tmp = Float64(0.16666666666666666 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 2e-7)
		tmp = -0.5;
	else
		tmp = 0.16666666666666666 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-7], -0.5, N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.9999999999999999e-7

   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 99.1%

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

   if 1.9999999999999999e-7 < (*.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. *-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 99.2

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

   5. Applied rewrites 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 2e-7) -0.5 (* x (* x 0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 2e-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 * x) <= 2d-7) 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) <= 2e-7) {
		tmp = -0.5;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 2e-7:
		tmp = -0.5
	else:
		tmp = x * (x * 0.16666666666666666)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 2e-7)
		tmp = -0.5;
	else
		tmp = Float64(x * Float64(x * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 2e-7)
		tmp = -0.5;
	else
		tmp = x * (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-7], -0.5, N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.9999999999999999e-7

   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 99.1%

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

   if 1.9999999999999999e-7 < (*.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. *-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 99.2

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

   5. Applied rewrites 99.2%

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

   7. Applied rewrites 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-7}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * 0.16666666666666666), $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 99.8

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

4. Applied rewrites 99.8%

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

Alternative 5: 50.4% 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 45.8%

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

Reproduce

herbie shell --seed 2024238 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))