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

Percentage Accurate: 99.8% → 99.9%

Time: 4.2s

Alternatives: 4

Speedup: 1.0×

• 25.2% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (x * 0.16666666666666666d0)) - 0.5d0
end function

Java

public static double code(double x) {
	return (x * (x * 0.16666666666666666)) - 0.5;
}

Python

def code(x):
	return (x * (x * 0.16666666666666666)) - 0.5

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x * (x * 0.16666666666666666)) - 0.5;
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. div-sub 99.6%

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

   2. div-inv 99.5%

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

   3. metadata-eval 99.5%

      \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{0.16666666666666666} - \frac{3}{6} \]

   4. associate-*l* 99.8%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot 0.16666666666666666\right)} - \frac{3}{6} \]

   5. metadata-eval 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

   \[\leadsto x \cdot \left(x \cdot 0.16666666666666666\right) - 0.5 \]

Alternative 2: 98.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 3.0], -0.5, N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $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. Taylor expanded in x around 0 98.5%

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

   if 3 < (*.f64 x x)

   1. Initial program 99.1%

      \[\frac{x \cdot x - 3}{6} \]

   2. Taylor expanded in x around inf 97.4%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 97.4%

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

   4. Simplified 97.4%

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

4. Final simplification 97.9%

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

Alternative 3: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.002:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 0.002) -0.5 (* x (/ x 6.0))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.002) {
		tmp = -0.5;
	} else {
		tmp = x * (x / 6.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 0.002d0) then
        tmp = -0.5d0
    else
        tmp = x * (x / 6.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.002) {
		tmp = -0.5;
	} else {
		tmp = x * (x / 6.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.002:
		tmp = -0.5
	else:
		tmp = x * (x / 6.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.002)
		tmp = -0.5;
	else
		tmp = Float64(x * Float64(x / 6.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.002)
		tmp = -0.5;
	else
		tmp = x * (x / 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.002], -0.5, N[(x * N[(x / 6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

      \[\frac{x \cdot x - 3}{6} \]

   2. Taylor expanded in x around 0 98.5%

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

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

   1. Initial program 99.1%

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

      1. sqr-neg 99.1%

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

      2. div-sub 99.2%

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

      3. *-lft-identity 99.2%

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

      4. metadata-eval 99.2%

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

      5. associate-*r/ 99.2%

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

      6. associate-/l* 99.0%

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

      7. associate-/r/ 99.1%

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

      8. *-commutative 99.1%

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

      9. fma-neg 99.1%

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

      10. sqr-neg 99.1%

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

      11. metadata-eval 99.1%

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

      12. metadata-eval 99.1%

          \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{0.16666666666666666}, -\frac{3}{6}\right) \]

      13. metadata-eval 99.1%

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

      14. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. metadata-eval 99.1%

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

      2. metadata-eval 99.1%

         \[\leadsto \mathsf{fma}\left(x \cdot x, 0.16666666666666666, -\color{blue}{\frac{3}{6}}\right) \]

      3. fma-neg 99.1%

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot 0.16666666666666666 - \frac{3}{6}} \]

      4. metadata-eval 99.1%

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

      5. div-inv 99.2%

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

      6. div-sub 99.1%

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

      7. clear-num 99.0%

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

      8. fma-neg 99.0%

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

      9. metadata-eval 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in x around inf 97.3%

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

      1. unpow2 97.3%

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

   8. Simplified 97.3%

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

   9. Taylor expanded in x around 0 97.4%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot {x}^{2}} \]
   10. Step-by-step derivation

       1. metadata-eval 97.4%

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

       2. unpow2 97.4%

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

       3. associate-/r/ 97.3%

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

       4. associate-/r* 97.8%

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

       5. metadata-eval 97.8%

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

       6. associate-*l/ 97.8%

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

       7. associate-/l* 97.9%

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

       8. rem-square-sqrt 40.9%

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

       9. associate-*l/ 40.8%

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

       10. associate-/r/ 40.9%

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

       11. remove-double-div 40.9%

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

       12. associate-*l/ 40.9%

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

       13. rem-square-sqrt 98.0%

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

   11. Simplified 98.0%

       \[\leadsto \color{blue}{x \cdot \frac{x}{6}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.002:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \]

Alternative 4: 50.1% accurate, 7.0× 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 99.6%

   \[\frac{x \cdot x - 3}{6} \]

2. Taylor expanded in x around 0 49.3%

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

3. Final simplification 49.3%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))