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

Percentage Accurate: 99.9% → 99.9%

Time: 3.3s

Alternatives: 5

Speedup: 1.0×

• 24.4% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, x, -3\right)}{6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fma x x -3.0) 6.0))

C

double code(double x) {
	return fma(x, x, -3.0) / 6.0;
}

Julia

function code(x)
	return Float64(fma(x, x, -3.0) / 6.0)
end

Wolfram

code[x_] := N[(N[(x * x + -3.0), $MachinePrecision] / 6.0), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. fma-neg 99.9%

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

   2. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.0004:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* x x) 0.0004) -0.5 (* (* x x) 0.16666666666666666)))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.0004) {
		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) <= 0.0004d0) 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) <= 0.0004) {
		tmp = -0.5;
	} else {
		tmp = (x * x) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.0004:
		tmp = -0.5
	else:
		tmp = (x * x) * 0.16666666666666666
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.0004)
		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) <= 0.0004)
		tmp = -0.5;
	else
		tmp = (x * x) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.0004], -0.5, N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.00000000000000019e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.3%

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

   if 4.00000000000000019e-4 < (*.f64 x x)

   1. Initial program 99.8%

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

   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 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.0004:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot 0.16666666666666666\\ \end{array} \]

Alternative 3: 98.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.0004:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (* x x) 0.0004) -0.5 (* x (/ x 6.0))))

C

double code(double x) {
	double tmp;
	if ((x * x) <= 0.0004) {
		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) <= 0.0004d0) 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) <= 0.0004) {
		tmp = -0.5;
	} else {
		tmp = x * (x / 6.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x * x) <= 0.0004:
		tmp = -0.5
	else:
		tmp = x * (x / 6.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(x * x) <= 0.0004)
		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) <= 0.0004)
		tmp = -0.5;
	else
		tmp = x * (x / 6.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(x * x), $MachinePrecision], 0.0004], -0.5, N[(x * N[(x / 6.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.00000000000000019e-4

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.3%

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

   if 4.00000000000000019e-4 < (*.f64 x x)

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 97.2%

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

      1. unpow2 97.2%

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

   4. Simplified 97.2%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{6} \]
   5. Step-by-step derivation

      1. associate-/l* 97.2%

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

      2. associate-/r/ 97.3%

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

   6. Applied egg-rr 97.3%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 0.0004:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{6}\\ \end{array} \]

Alternative 4: 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]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

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 99.9%

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

2. Taylor expanded in x around 0 53.2%

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

3. Final simplification 53.2%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023221 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))