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

Percentage Accurate: 99.9% → 99.9%

Time: 6.5s

Alternatives: 8

Speedup: 1.0×

• 25.1% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 - N[(N[(x * N[(x * 0.12), $MachinePrecision]), $MachinePrecision] + 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. Step-by-step derivation

   1. +-commutative 99.9%

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

   2. distribute-lft-in 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto 1 - \left(x \cdot \left(x \cdot 0.12\right) + x \cdot 0.253\right) \]

Alternative 2: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

def code(x):
	tmp = 0
	if (x <= -4.2) 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 <= -4.2) || !(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 <= -4.2) || ~((x <= 2.0)))
		tmp = -0.12 * (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.2], 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 < -4.20000000000000018 or 2 < x

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

   2. Taylor expanded in x around inf 97.3%

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

      1. unpow2 97.3%

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

   4. Simplified 97.3%

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

   if -4.20000000000000018 < x < 2

   1. Initial program 100.0%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

   2. Taylor expanded in x around 0 96.4%

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

4. Final simplification 96.9%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

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 = -0.12 * (x * x);
	}
	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 = (-0.12d0) * (x * x)
    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 = -0.12 * (x * x);
	}
	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 = -0.12 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(x * Float64(x * -0.12));
	elseif (x <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(-0.12 * Float64(x * x));
	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 = -0.12 * (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000018

   1. Initial program 99.7%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

   2. Taylor expanded in x around inf 95.5%

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

      1. unpow2 95.5%

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

      2. *-commutative 95.5%

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

      3. associate-*l* 96.5%

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

   4. Simplified 96.5%

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

   if -4.20000000000000018 < x < 2

   1. Initial program 100.0%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

   2. Taylor expanded in x around 0 96.4%

      \[\leadsto \color{blue}{1} \]

   if 2 < x

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

   2. Taylor expanded in x around inf 99.8%

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

      1. unpow2 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 97.2%

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

Alternative 4: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

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 = -0.12 * (x * x);
	}
	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 = (-0.12d0) * (x * x)
    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 = -0.12 * (x * x);
	}
	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 = -0.12 * (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(x * Float64(x * -0.12));
	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 <= -4.2)
		tmp = x * (x * -0.12);
	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, -4.2], N[(x * N[(x * -0.12), $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 < -4.20000000000000018

   1. Initial program 99.7%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

   2. Taylor expanded in x around inf 95.5%

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

      1. unpow2 95.5%

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

      2. *-commutative 95.5%

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

      3. associate-*l* 96.5%

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

   4. Simplified 96.5%

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

   if -4.20000000000000018 < x < 2

   1. Initial program 100.0%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

   2. Taylor expanded in x around 0 98.0%

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

      1. *-commutative 98.0%

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

   4. Simplified 98.0%

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

   if 2 < x

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

   2. Taylor expanded in x around inf 99.8%

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

      1. unpow2 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 98.0%

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

Alternative 5: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x * (-0.253 + (x * -0.12));
	} 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 <= (-4.2d0)) then
        tmp = x * ((-0.253d0) + (x * (-0.12d0)))
    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 <= -4.2) {
		tmp = x * (-0.253 + (x * -0.12));
	} 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 <= -4.2:
		tmp = x * (-0.253 + (x * -0.12))
	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 <= -4.2)
		tmp = Float64(x * Float64(-0.253 + Float64(x * -0.12)));
	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 <= -4.2)
		tmp = x * (-0.253 + (x * -0.12));
	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, -4.2], N[(x * N[(-0.253 + N[(x * -0.12), $MachinePrecision]), $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 < -4.20000000000000018

   1. Initial program 99.7%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

   2. Taylor expanded in x around inf 96.6%

      \[\leadsto \color{blue}{-0.253 \cdot x + -0.12 \cdot {x}^{2}} \]
   3. Step-by-step derivation

      1. unpow2 96.6%

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

      2. associate-*r* 97.7%

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

      3. distribute-rgt-out 97.7%

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

      4. *-commutative 97.7%

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

   4. Simplified 97.7%

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

   if -4.20000000000000018 < x < 2

   1. Initial program 100.0%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

   2. Taylor expanded in x around 0 98.0%

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

      1. *-commutative 98.0%

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

   4. Simplified 98.0%

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

   if 2 < x

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

   2. Taylor expanded in x around inf 99.8%

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

      1. unpow2 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 98.3%

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

Alternative 6: 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. Final simplification 99.9%

   \[\leadsto 1 - x \cdot \left(x \cdot 0.12 + 0.253\right) \]

Alternative 7: 97.9% 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. Step-by-step derivation

   1. flip-+ 99.8%

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

   2. associate-*r/ 90.2%

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

   3. metadata-eval 90.2%

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

   4. swap-sqr 90.2%

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

   5. metadata-eval 90.2%

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

   6. *-commutative 90.2%

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

   7. cancel-sign-sub-inv 90.2%

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

   8. metadata-eval 90.2%

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

3. Applied egg-rr 90.2%

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

   1. associate-/l* 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*l* 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around inf 97.1%

   \[\leadsto 1 - \frac{x}{\color{blue}{\frac{8.333333333333334}{x}}} \]

7. Final simplification 97.1%

   \[\leadsto 1 - \frac{x}{\frac{8.333333333333334}{x}} \]

Alternative 8: 49.6% 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. Taylor expanded in x around 0 48.7%

   \[\leadsto \color{blue}{1} \]

3. Final simplification 48.7%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))