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

Percentage Accurate: 99.9% → 99.8%

Time: 7.7s

Alternatives: 8

Speedup: 1.0×

• 24.4% 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: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. distribute-rgt-in 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

   4. associate-*l* 99.9%

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

   5. pow2 99.9%

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

4. Applied egg-rr 99.9%

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

   1. unpow2 99.9%

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

6. Applied egg-rr 99.9%

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

Alternative 2: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.0)) {
		tmp = x * ((x * -0.12) - 0.253);
	} 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)) - 0.253d0)
    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) - 0.253);
	} 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) - 0.253)
	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(x * Float64(Float64(x * -0.12) - 0.253));
	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) - 0.253);
	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[(x * N[(N[(x * -0.12), $MachinePrecision] - 0.253), $MachinePrecision]), $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 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-rgt-in 76.8%

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

      3. distribute-neg-in 76.8%

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

      4. distribute-lft-neg-in 76.8%

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

      5. metadata-eval 76.8%

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

      6. unpow2 76.8%

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

      7. associate-*r* 76.8%

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

      8. unpow2 76.8%

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

      9. associate-*r* 99.3%

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

      10. distribute-lft-neg-in 99.3%

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

      11. associate-*l* 99.3%

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

      12. lft-mult-inverse 99.3%

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

      13. metadata-eval 99.3%

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

      14. metadata-eval 99.3%

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

      15. distribute-rgt-in 99.3%

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

      16. *-commutative 99.3%

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

      17. fma-define 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around 0 99.3%

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

   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 98.7%

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

      1. *-commutative 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 99.0%

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

Alternative 3: 98.6% accurate, 0.6× speedup

Math

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

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(x * Float64(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[(x * N[(x * -0.12), $MachinePrecision]), $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 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-rgt-in 76.8%

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

      3. distribute-neg-in 76.8%

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

      4. distribute-lft-neg-in 76.8%

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

      5. metadata-eval 76.8%

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

      6. unpow2 76.8%

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

      7. associate-*r* 76.8%

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

      8. unpow2 76.8%

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

      9. associate-*r* 99.3%

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

      10. distribute-lft-neg-in 99.3%

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

      11. associate-*l* 99.3%

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

      12. lft-mult-inverse 99.3%

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

      13. metadata-eval 99.3%

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

      14. metadata-eval 99.3%

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

      15. distribute-rgt-in 99.3%

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

      16. *-commutative 99.3%

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

      17. fma-define 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf 98.1%

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

      1. *-commutative 98.1%

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

   8. Simplified 98.1%

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

   if -4.20000000000000018 < x < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.7%

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

      1. *-commutative 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 98.4%

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

Alternative 4: 98.2% accurate, 0.6× speedup

Math

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

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(x * Float64(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[(x * N[(x * -0.12), $MachinePrecision]), $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 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-rgt-in 76.8%

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

      3. distribute-neg-in 76.8%

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

      4. distribute-lft-neg-in 76.8%

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

      5. metadata-eval 76.8%

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

      6. unpow2 76.8%

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

      7. associate-*r* 76.8%

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

      8. unpow2 76.8%

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

      9. associate-*r* 99.3%

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

      10. distribute-lft-neg-in 99.3%

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

      11. associate-*l* 99.3%

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

      12. lft-mult-inverse 99.3%

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

      13. metadata-eval 99.3%

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

      14. metadata-eval 99.3%

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

      15. distribute-rgt-in 99.3%

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

      16. *-commutative 99.3%

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

      17. fma-define 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in x around inf 98.1%

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

      1. *-commutative 98.1%

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

   8. Simplified 98.1%

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

   if -4.20000000000000018 < x < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.6%

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

4. Final simplification 97.9%

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

Alternative 5: 99.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 99.9%

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

   1. *-un-lft-identity 99.9%

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

   2. un-div-inv 99.9%

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

5. Applied egg-rr 99.9%

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

   1. *-lft-identity 99.9%

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

7. Simplified 99.9%

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

Alternative 6: 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 99.8%

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

Alternative 7: 51.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.253\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 2

   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 66.6%

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

   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 99.7%

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

      1. mul-1-neg 99.7%

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

      2. distribute-rgt-in 99.6%

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

      3. distribute-neg-in 99.6%

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

      4. distribute-lft-neg-in 99.6%

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

      5. metadata-eval 99.6%

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

      6. unpow2 99.6%

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

      7. associate-*r* 99.6%

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

      8. unpow2 99.6%

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

      9. associate-*r* 99.6%

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

      10. distribute-lft-neg-in 99.6%

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

      11. associate-*l* 99.6%

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

      12. lft-mult-inverse 99.6%

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

      13. metadata-eval 99.6%

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

      14. metadata-eval 99.6%

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

      15. distribute-rgt-in 99.6%

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

      16. *-commutative 99.6%

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

      17. fma-define 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 7.2%

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

      1. *-commutative 7.2%

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

   8. Simplified 7.2%

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

Alternative 8: 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 99.8%

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

3. Taylor expanded in x around 0 48.9%

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

Reproduce

herbie shell --seed 2024177 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))