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

Percentage Accurate: 99.9% → 99.9%

Time: 6.7s

Alternatives: 6

Speedup: 1.0×

• 25.4% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. distribute-rgt-in 99.9%

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

   3. *-commutative 99.9%

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

   4. fma-define 99.9%

      \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(x \cdot 0.12, x, x \cdot 0.253\right)} \]

4. Applied egg-rr 99.9%

   \[\leadsto 1 - \color{blue}{\mathsf{fma}\left(x \cdot 0.12, x, x \cdot 0.253\right)} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.1) (not (<= x 2.0))) (* -0.12 (* x x)) (- 1.0 (* x 0.253))))

C

double code(double x) {
	double tmp;
	if ((x <= -4.1) || !(x <= 2.0)) {
		tmp = -0.12 * (x * x);
	} 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 <= (-4.1d0)) .or. (.not. (x <= 2.0d0))) then
        tmp = (-0.12d0) * (x * x)
    else
        tmp = 1.0d0 - (x * 0.253d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -4.1) || !(x <= 2.0)) {
		tmp = -0.12 * (x * x);
	} else {
		tmp = 1.0 - (x * 0.253);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.1) or not (x <= 2.0):
		tmp = -0.12 * (x * x)
	else:
		tmp = 1.0 - (x * 0.253)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.1) || !(x <= 2.0))
		tmp = Float64(-0.12 * Float64(x * x));
	else
		tmp = Float64(1.0 - Float64(x * 0.253));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -4.1) || ~((x <= 2.0)))
		tmp = -0.12 * (x * x);
	else
		tmp = 1.0 - (x * 0.253);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.1], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(-0.12 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.0999999999999996 or 2 < 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 98.3%

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

      1. *-commutative 98.3%

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

   5. Simplified 98.3%

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

      1. unpow2 99.8%

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

   7. Applied egg-rr 98.3%

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

   if -4.0999999999999996 < x < 2

   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 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.1) (not (<= x 2.0))) (* -0.12 (* x x)) 1.0))

C

double code(double x) {
	double tmp;
	if ((x <= -4.1) || !(x <= 2.0)) {
		tmp = -0.12 * (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4.1d0)) .or. (.not. (x <= 2.0d0))) then
        tmp = (-0.12d0) * (x * x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -4.1) || !(x <= 2.0)) {
		tmp = -0.12 * (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.1) or not (x <= 2.0):
		tmp = -0.12 * (x * x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.1) || !(x <= 2.0))
		tmp = Float64(-0.12 * Float64(x * x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -4.1) || ~((x <= 2.0)))
		tmp = -0.12 * (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.1], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(-0.12 * N[(x * x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.0999999999999996 or 2 < 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 98.3%

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

      1. *-commutative 98.3%

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

   5. Simplified 98.3%

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

      1. unpow2 99.8%

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

   7. Applied egg-rr 98.3%

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

   if -4.0999999999999996 < x < 2

   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 98.4%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;x \cdot \left(x \cdot -0.12 - 0.253\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.1)
   (* x (- (* x -0.12) 0.253))
   (if (<= x 2.0) (- 1.0 (* x 0.253)) (* -0.12 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.1) {
		tmp = x * ((x * -0.12) - 0.253);
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = -0.12 * (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.1d0)) then
        tmp = x * ((x * (-0.12d0)) - 0.253d0)
    else if (x <= 2.0d0) then
        tmp = 1.0d0 - (x * 0.253d0)
    else
        tmp = (-0.12d0) * (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.1) {
		tmp = x * ((x * -0.12) - 0.253);
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = -0.12 * (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.1:
		tmp = x * ((x * -0.12) - 0.253)
	elif x <= 2.0:
		tmp = 1.0 - (x * 0.253)
	else:
		tmp = -0.12 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.1)
		tmp = Float64(x * Float64(Float64(x * -0.12) - 0.253));
	elseif (x <= 2.0)
		tmp = Float64(1.0 - Float64(x * 0.253));
	else
		tmp = Float64(-0.12 * Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.1)
		tmp = x * ((x * -0.12) - 0.253);
	elseif (x <= 2.0)
		tmp = 1.0 - (x * 0.253);
	else
		tmp = -0.12 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.1], N[(x * N[(N[(x * -0.12), $MachinePrecision] - 0.253), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.0], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision], N[(-0.12 * N[(x * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0999999999999996

   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 99.8%

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(-1 \cdot \frac{0.253 - \frac{1}{x}}{x} - 0.12\right)} \]
   4. Step-by-step derivation

      1. unpow2 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around inf 98.5%

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

   7. Taylor expanded in x around 0 98.6%

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

   if -4.0999999999999996 < x < 2

   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 99.1%

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

      1. *-commutative 99.1%

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

   5. Simplified 99.1%

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

   if 2 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. *-commutative 99.8%

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

   5. Simplified 99.8%

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

      1. unpow2 99.8%

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

   7. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\left(x \cdot x\right)} \cdot -0.12 \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;x \cdot \left(x \cdot -0.12 - 0.253\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 6: 49.2% 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 49.6%

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

Reproduce

herbie shell --seed 2024135 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))