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

Percentage Accurate: 99.9% → 99.9%

Time: 8.7s

Alternatives: 7

Speedup: 1.0×

• 23.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-define 99.9%

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

3. Simplified 99.9%

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

Alternative 2: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.2) (not (<= x 2.0))) (* (* x x) -0.12) (- 1.0 (* x 0.253))))

C

double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.0)) {
		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 <= 2.0d0))) 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 <= 2.0)) {
		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 <= 2.0):
		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 <= 2.0))
		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 <= 2.0)))
		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, 2.0]], $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 < 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 97.8%

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

      1. *-commutative 97.8%

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

   5. Simplified 97.8%

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

      1. unpow2 97.8%

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

   7. Applied egg-rr 97.8%

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

   if -4.20000000000000018 < 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 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.2) (not (<= x 2.0))) (* (* x x) -0.12) 1.0))

C

double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.0)) {
		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 <= 2.0d0))) 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 <= 2.0)) {
		tmp = (x * x) * -0.12;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.2) or not (x <= 2.0):
		tmp = (x * x) * -0.12
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.2) || !(x <= 2.0))
		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 <= 2.0)))
		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, 2.0]], $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 < 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 97.8%

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

      1. *-commutative 97.8%

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

   5. Simplified 97.8%

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

      1. unpow2 97.8%

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

   7. Applied egg-rr 97.8%

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

   if -4.20000000000000018 < 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 99.5%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 5: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.0144 \cdot \left(x \cdot 8.333333333333334\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- 1.0 (* x (* 0.0144 (* x 8.333333333333334)))))

C

double code(double x) {
	return 1.0 - (x * (0.0144 * (x * 8.333333333333334)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.0144d0 * (x * 8.333333333333334d0)))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (0.0144 * (x * 8.333333333333334)));
}

Python

def code(x):
	return 1.0 - (x * (0.0144 * (x * 8.333333333333334)))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.0144 * Float64(x * 8.333333333333334))))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.0144 * (x * 8.333333333333334)));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.0144 * N[(x * 8.333333333333334), $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. 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. Step-by-step derivation

   1. flip-+ 76.5%

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

   2. associate-*r/ 76.5%

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

   3. metadata-eval 76.5%

      \[\leadsto 1 - x \cdot \frac{x \cdot \left(\color{blue}{0.0144} - \left(0.253 \cdot \frac{1}{x}\right) \cdot \left(0.253 \cdot \frac{1}{x}\right)\right)}{0.12 - 0.253 \cdot \frac{1}{x}} \]

   4. un-div-inv 76.5%

      \[\leadsto 1 - x \cdot \frac{x \cdot \left(0.0144 - \color{blue}{\frac{0.253}{x}} \cdot \left(0.253 \cdot \frac{1}{x}\right)\right)}{0.12 - 0.253 \cdot \frac{1}{x}} \]

   5. un-div-inv 76.5%

      \[\leadsto 1 - x \cdot \frac{x \cdot \left(0.0144 - \frac{0.253}{x} \cdot \color{blue}{\frac{0.253}{x}}\right)}{0.12 - 0.253 \cdot \frac{1}{x}} \]

   6. frac-times 76.5%

      \[\leadsto 1 - x \cdot \frac{x \cdot \left(0.0144 - \color{blue}{\frac{0.253 \cdot 0.253}{x \cdot x}}\right)}{0.12 - 0.253 \cdot \frac{1}{x}} \]

   7. metadata-eval 76.5%

      \[\leadsto 1 - x \cdot \frac{x \cdot \left(0.0144 - \frac{\color{blue}{0.064009}}{x \cdot x}\right)}{0.12 - 0.253 \cdot \frac{1}{x}} \]

   8. unpow2 76.5%

      \[\leadsto 1 - x \cdot \frac{x \cdot \left(0.0144 - \frac{0.064009}{\color{blue}{{x}^{2}}}\right)}{0.12 - 0.253 \cdot \frac{1}{x}} \]

   9. un-div-inv 76.5%

      \[\leadsto 1 - x \cdot \frac{x \cdot \left(0.0144 - \frac{0.064009}{{x}^{2}}\right)}{0.12 - \color{blue}{\frac{0.253}{x}}} \]

5. Applied egg-rr 76.5%

   \[\leadsto 1 - x \cdot \color{blue}{\frac{x \cdot \left(0.0144 - \frac{0.064009}{{x}^{2}}\right)}{0.12 - \frac{0.253}{x}}} \]
6. Step-by-step derivation

   1. *-commutative 76.5%

      \[\leadsto 1 - x \cdot \frac{\color{blue}{\left(0.0144 - \frac{0.064009}{{x}^{2}}\right) \cdot x}}{0.12 - \frac{0.253}{x}} \]

   2. associate-/l* 76.1%

      \[\leadsto 1 - x \cdot \color{blue}{\left(\left(0.0144 - \frac{0.064009}{{x}^{2}}\right) \cdot \frac{x}{0.12 - \frac{0.253}{x}}\right)} \]

7. Simplified 76.1%

   \[\leadsto 1 - x \cdot \color{blue}{\left(\left(0.0144 - \frac{0.064009}{{x}^{2}}\right) \cdot \frac{x}{0.12 - \frac{0.253}{x}}\right)} \]

8. Taylor expanded in x around inf 99.0%

   \[\leadsto 1 - x \cdot \left(\color{blue}{0.0144} \cdot \frac{x}{0.12 - \frac{0.253}{x}}\right) \]

9. Taylor expanded in x around inf 98.7%

   \[\leadsto 1 - x \cdot \left(0.0144 \cdot \color{blue}{\left(8.333333333333334 \cdot x\right)}\right) \]
10. Step-by-step derivation

    1. *-commutative 98.7%

       \[\leadsto 1 - x \cdot \left(0.0144 \cdot \color{blue}{\left(x \cdot 8.333333333333334\right)}\right) \]

11. Simplified 98.7%

    \[\leadsto 1 - x \cdot \left(0.0144 \cdot \color{blue}{\left(x \cdot 8.333333333333334\right)}\right) \]
12. Add Preprocessing

Alternative 6: 97.8% 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 98.6%

   \[\leadsto 1 - x \cdot \color{blue}{\left(0.12 \cdot x\right)} \]
5. Step-by-step derivation

   1. *-commutative 98.6%

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

6. Simplified 98.6%

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

Alternative 7: 50.1% 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 52.1%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))