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

Percentage Accurate: 99.9% → 99.9%

Time: 11.1s

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, 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: 98.6% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2)
   (* x (- -0.253 (* 0.12 x)))
   (if (<= x 2.0) (- 1.0 (* x 0.253)) (* x (* x -0.12)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x * (-0.253 - (0.12 * x));
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.2d0)) then
        tmp = x * ((-0.253d0) - (0.12d0 * x))
    else if (x <= 2.0d0) then
        tmp = 1.0d0 - (x * 0.253d0)
    else
        tmp = x * (x * (-0.12d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x * (-0.253 - (0.12 * x));
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = x * (-0.253 - (0.12 * x))
	elif x <= 2.0:
		tmp = 1.0 - (x * 0.253)
	else:
		tmp = x * (x * -0.12)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(x * Float64(-0.253 - Float64(0.12 * x)));
	elseif (x <= 2.0)
		tmp = Float64(1.0 - Float64(x * 0.253));
	else
		tmp = Float64(x * Float64(x * -0.12));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = x * (-0.253 - (0.12 * x));
	elseif (x <= 2.0)
		tmp = 1.0 - (x * 0.253);
	else
		tmp = x * (x * -0.12);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(x * N[(-0.253 - N[(0.12 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.0], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000018

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   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 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 2 < x

   1. Initial program 99.8%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

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

Alternative 3: 98.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2)
   (* (* x x) -0.12)
   (if (<= x 2.0) (- 1.0 (* x 0.253)) (* x (* x -0.12)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = (x * x) * -0.12;
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.2d0)) then
        tmp = (x * x) * (-0.12d0)
    else if (x <= 2.0d0) then
        tmp = 1.0d0 - (x * 0.253d0)
    else
        tmp = x * (x * (-0.12d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = (x * x) * -0.12;
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = (x * x) * -0.12
	elif x <= 2.0:
		tmp = 1.0 - (x * 0.253)
	else:
		tmp = x * (x * -0.12)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(Float64(x * x) * -0.12);
	elseif (x <= 2.0)
		tmp = Float64(1.0 - Float64(x * 0.253));
	else
		tmp = Float64(x * Float64(x * -0.12));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = (x * x) * -0.12;
	elseif (x <= 2.0)
		tmp = 1.0 - (x * 0.253);
	else
		tmp = x * (x * -0.12);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(N[(x * x), $MachinePrecision] * -0.12), $MachinePrecision], If[LessEqual[x, 2.0], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000018

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   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 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 2 < x

   1. Initial program 99.8%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

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

Alternative 4: 97.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2) (* (* x x) -0.12) (if (<= x 2.0) 1.0 (* x (* x -0.12)))))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.2d0)) then
        tmp = (x * x) * (-0.12d0)
    else if (x <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x * (x * (-0.12d0))
    end if
    code = tmp
end function

Java

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

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = (x * x) * -0.12
	elif x <= 2.0:
		tmp = 1.0
	else:
		tmp = x * (x * -0.12)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(Float64(x * x) * -0.12);
	elseif (x <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * -0.12));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = (x * x) * -0.12;
	elseif (x <= 2.0)
		tmp = 1.0;
	else
		tmp = x * (x * -0.12);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(N[(x * x), $MachinePrecision] * -0.12), $MachinePrecision], If[LessEqual[x, 2.0], 1.0, N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000018

   1. Initial program 99.6%

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

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   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 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 2 < x

   1. Initial program 99.8%

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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

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

Alternative 5: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot -0.12\right)\\ \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x -0.12)))) (if (<= x -4.2) t_0 (if (<= x 2.0) 1.0 t_0))))

C

double code(double x) {
	double t_0 = x * (x * -0.12);
	double tmp;
	if (x <= -4.2) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (x * (-0.12d0))
    if (x <= (-4.2d0)) then
        tmp = t_0
    else if (x <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (x * -0.12);
	double tmp;
	if (x <= -4.2) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * -0.12)
	tmp = 0
	if x <= -4.2:
		tmp = t_0
	elif x <= 2.0:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * -0.12))
	tmp = 0.0
	if (x <= -4.2)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (x * -0.12);
	tmp = 0.0;
	if (x <= -4.2)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.2], t$95$0, If[LessEqual[x, 2.0], 1.0, t$95$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

   4. Applied egg-rr 0

      \[\leadsto expr\]

   5. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   7. Simplified 0

      \[\leadsto expr\]

   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 0

      \[\leadsto expr\]

   5. Simplified 0

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

Alternative 6: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 - \frac{x}{\frac{8.333333333333334}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (/ x (/ 8.333333333333334 x))))

C

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

Fortran

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

Java

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

Python

def code(x):
	return 1.0 - (x / (8.333333333333334 / x))

Julia

function code(x)
	return Float64(1.0 - Float64(x / Float64(8.333333333333334 / x)))
end

MATLAB

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

Wolfram

code[x_] := N[(1.0 - N[(x / N[(8.333333333333334 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in x around inf 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]
8. 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 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]
6. 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)))))