Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

Alternatives: 5

Speedup: 1.0×

• 25.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))

C

double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.253d0 + (x * 0.12d0)))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Python

def code(x):
	return 1.0 - (x * (0.253 + (x * 0.12)))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.253 + (x * 0.12)));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))

C

double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.253d0 + (x * 0.12d0)))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Python

def code(x):
	return 1.0 - (x * (0.253 + (x * 0.12)))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.253 + (x * 0.12)));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))

C

double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.253d0 + (x * 0.12d0)))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

Python

def code(x):
	return 1.0 - (x * (0.253 + (x * 0.12)))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.253 + (x * 0.12)));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2.1\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.2) (not (<= x 2.1))) (* (* x x) -0.12) (- 1.0 (* x 0.253))))

C

double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.1)) {
		tmp = (x * x) * -0.12;
	} else {
		tmp = 1.0 - (x * 0.253);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4.2d0)) .or. (.not. (x <= 2.1d0))) then
        tmp = (x * x) * (-0.12d0)
    else
        tmp = 1.0d0 - (x * 0.253d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.1)) {
		tmp = (x * x) * -0.12;
	} else {
		tmp = 1.0 - (x * 0.253);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.2) or not (x <= 2.1):
		tmp = (x * x) * -0.12
	else:
		tmp = 1.0 - (x * 0.253)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.2) || !(x <= 2.1))
		tmp = Float64(Float64(x * x) * -0.12);
	else
		tmp = Float64(1.0 - Float64(x * 0.253));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -4.2) || ~((x <= 2.1)))
		tmp = (x * x) * -0.12;
	else
		tmp = 1.0 - (x * 0.253);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.2], N[Not[LessEqual[x, 2.1]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] * -0.12), $MachinePrecision], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018 or 2.10000000000000009 < x

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.2%

      \[\leadsto \color{blue}{-0.12 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 98.2%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]
   6. Step-by-step derivation

      1. unpow2 98.2%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot -0.12 \]

   7. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot -0.12 \]

   if -4.20000000000000018 < x < 2.10000000000000009

   1. Initial program 100.0%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.5%

      \[\leadsto 1 - \color{blue}{0.253 \cdot x} \]
   4. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto 1 - \color{blue}{x \cdot 0.253} \]

   5. Simplified 98.5%

      \[\leadsto 1 - \color{blue}{x \cdot 0.253} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2.1\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2.1\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.2) (not (<= x 2.1))) (* (* x x) -0.12) 1.0))

C

double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.1)) {
		tmp = (x * x) * -0.12;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4.2d0)) .or. (.not. (x <= 2.1d0))) then
        tmp = (x * x) * (-0.12d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.1)) {
		tmp = (x * x) * -0.12;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.2) or not (x <= 2.1):
		tmp = (x * x) * -0.12
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.2) || !(x <= 2.1))
		tmp = Float64(Float64(x * x) * -0.12);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -4.2) || ~((x <= 2.1)))
		tmp = (x * x) * -0.12;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.2], N[Not[LessEqual[x, 2.1]], $MachinePrecision]], N[(N[(x * x), $MachinePrecision] * -0.12), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018 or 2.10000000000000009 < x

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.2%

      \[\leadsto \color{blue}{-0.12 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 98.2%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]
   6. Step-by-step derivation

      1. unpow2 98.2%

         \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot -0.12 \]

   7. Applied egg-rr 98.2%

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot -0.12 \]

   if -4.20000000000000018 < x < 2.10000000000000009

   1. Initial program 100.0%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 97.4%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2.1\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(x \cdot 0.12\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (* x 0.12))))

C

double code(double x) {
	return 1.0 - (x * (x * 0.12));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (x * 0.12d0))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (x * 0.12));
}

Python

def code(x):
	return 1.0 - (x * (x * 0.12))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(x * 0.12)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (x * 0.12));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 99.9%

   \[\leadsto 1 - x \cdot \color{blue}{\left(x \cdot \left(0.12 + 0.253 \cdot \frac{1}{x}\right)\right)} \]

4. Taylor expanded in x around inf 97.7%

   \[\leadsto 1 - x \cdot \left(x \cdot \color{blue}{0.12}\right) \]
5. Add Preprocessing

Alternative 5: 49.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 54.8%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024180 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))