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

Percentage Accurate: 99.8% → 99.9%

Time: 3.7s

Alternatives: 5

Speedup: 1.0×

• 24.7% 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 99.8%

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

   1. clear-num 99.8%

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

   2. inv-pow 99.8%

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

   3. fma-neg 99.8%

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

   4. metadata-eval 99.8%

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

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{{\left(\frac{6}{\mathsf{fma}\left(x, x, -3\right)}\right)}^{-1}} \]
4. Step-by-step derivation

   1. unpow-1 99.8%

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

   2. associate-/r/ 99.8%

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

   3. metadata-eval 99.8%

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

   4. fma-udef 99.8%

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

   5. distribute-rgt-in 99.8%

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

   6. associate-*l* 99.9%

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

   7. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 98.6% 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 98.7%

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

   if 3 < (*.f64 x x)

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 97.1%

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

      1. unpow2 97.1%

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

   4. Simplified 97.1%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot x\right)} \]
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: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.02:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.02) -0.5 (* x (* x 0.16666666666666666))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.02) {
		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) <= 0.02d0) 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) <= 0.02) {
		tmp = -0.5;
	} else {
		tmp = x * (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.02:
		tmp = -0.5
	else:
		tmp = x * (x * 0.16666666666666666)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.02)
		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) <= 0.02)
		tmp = -0.5;
	else
		tmp = x * (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.02], -0.5, N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.0200000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

   if 0.0200000000000000004 < (*.f64 x x)

   1. Initial program 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. fma-neg 99.7%

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

      4. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around inf 97.1%

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

      1. unpow2 97.1%

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

      2. *-commutative 97.1%

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

      3. associate-*r* 97.2%

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

   6. Simplified 97.2%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.02:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \]

Alternative 4: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.02:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{6}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 0.02) -0.5 (/ x (/ 6.0 x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.02) {
		tmp = -0.5;
	} else {
		tmp = x / (6.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x * x) <= 0.02d0) then
        tmp = -0.5d0
    else
        tmp = x / (6.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x * x) <= 0.02) {
		tmp = -0.5;
	} else {
		tmp = x / (6.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.02:
		tmp = -0.5
	else:
		tmp = x / (6.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.02)
		tmp = -0.5;
	else
		tmp = Float64(x / Float64(6.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 0.02)
		tmp = -0.5;
	else
		tmp = x / (6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.02], -0.5, N[(x / N[(6.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 0.0200000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.7%

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

   if 0.0200000000000000004 < (*.f64 x x)

   1. Initial program 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

      3. fma-neg 99.7%

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

      4. metadata-eval 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around inf 97.1%

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

      1. unpow2 97.1%

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

      2. *-commutative 97.1%

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

      3. associate-*r* 97.2%

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

   6. Simplified 97.2%

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

      1. associate-*r* 97.1%

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

      2. metadata-eval 97.1%

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

      3. div-inv 97.3%

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

      4. associate-/l* 97.4%

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

   8. Applied egg-rr 97.4%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.02:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{6}{x}}\\ \end{array} \]

Alternative 5: 50.7% 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 99.8%

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

2. Taylor expanded in x around 0 49.0%

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

3. Final simplification 49.0%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))