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

Percentage Accurate: 99.8% → 99.9%

Time: 4.2s

Alternatives: 6

Speedup: 1.0×

• 23.7% 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

\[\begin{array}{l} \\ \frac{x}{\frac{6}{x}} + -0.5 \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x / (6.0d0 / x)) + (-0.5d0)
end function

Java

public static double code(double x) {
	return (x / (6.0 / x)) + -0.5;
}

Python

def code(x):
	return (x / (6.0 / x)) + -0.5

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x / (6.0 / x)) + -0.5;
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]
7. Add Preprocessing

Alternative 2: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-15}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{6}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 5e-15) -0.5 (/ x (/ 6.0 x))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-15) {
		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) <= 5d-15) 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) <= 5e-15) {
		tmp = -0.5;
	} else {
		tmp = x / (6.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 5e-15:
		tmp = -0.5
	else:
		tmp = x / (6.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-15)
		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) <= 5e-15)
		tmp = -0.5;
	else
		tmp = x / (6.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-15], -0.5, N[(x / N[(6.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999999e-15

   1. Initial program 100.0%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 4.99999999999999999e-15 < (*.f64 x x)

   1. Initial program 99.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-15}:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{6} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 5e-15) -0.5 (* (/ x 6.0) x)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 5e-15) {
		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) <= 5d-15) 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) <= 5e-15) {
		tmp = -0.5;
	} else {
		tmp = (x / 6.0) * x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 5e-15:
		tmp = -0.5
	else:
		tmp = (x / 6.0) * x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 5e-15)
		tmp = -0.5;
	else
		tmp = Float64(Float64(x / 6.0) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x * x) <= 5e-15)
		tmp = -0.5;
	else
		tmp = (x / 6.0) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-15], -0.5, N[(N[(x / 6.0), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999999e-15

   1. Initial program 100.0%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 4.99999999999999999e-15 < (*.f64 x x)

   1. Initial program 99.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   9. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.7% accurate, 0.6× 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%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   if 3 < (*.f64 x x)

   1. Initial program 99.8%

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

   3. Simplified 0

      \[\leadsto expr\]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{6} \cdot x + -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (* (/ x 6.0) x) -0.5))

C

double code(double x) {
	return ((x / 6.0) * x) + -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x / 6.0d0) * x) + (-0.5d0)
end function

Java

public static double code(double x) {
	return ((x / 6.0) * x) + -0.5;
}

Python

def code(x):
	return ((x / 6.0) * x) + -0.5

Julia

function code(x)
	return Float64(Float64(Float64(x / 6.0) * x) + -0.5)
end

MATLAB

function tmp = code(x)
	tmp = ((x / 6.0) * x) + -0.5;
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

6. Applied egg-rr 0

   \[\leadsto expr\]
7. Add Preprocessing

Alternative 6: 50.6% 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 99.9%

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

3. Simplified 0

   \[\leadsto expr\]
4. Add Preprocessing

5. Taylor expanded in x around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))