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

Percentage Accurate: 99.9% → 99.9%

Time: 7.6s

Alternatives: 7

Speedup: 1.0×

• 25.0% 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: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(-0.253 + x \cdot -0.12\right)\\ \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (+ -0.253 (* x -0.12)))))
   (if (<= x -4.1) t_0 (if (<= x 2.0) (- 1.0 (* x 0.253)) t_0))))

C

double code(double x) {
	double t_0 = x * (-0.253 + (x * -0.12));
	double tmp;
	if (x <= -4.1) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} 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 * ((-0.253d0) + (x * (-0.12d0)))
    if (x <= (-4.1d0)) then
        tmp = t_0
    else if (x <= 2.0d0) then
        tmp = 1.0d0 - (x * 0.253d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x * (-0.253 + (x * -0.12));
	double tmp;
	if (x <= -4.1) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (-0.253 + (x * -0.12))
	tmp = 0
	if x <= -4.1:
		tmp = t_0
	elif x <= 2.0:
		tmp = 1.0 - (x * 0.253)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(-0.253 + Float64(x * -0.12)))
	tmp = 0.0
	if (x <= -4.1)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = Float64(1.0 - Float64(x * 0.253));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x * (-0.253 + (x * -0.12));
	tmp = 0.0;
	if (x <= -4.1)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = 1.0 - (x * 0.253);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(-0.253 + N[(x * -0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.1], t$95$0, If[LessEqual[x, 2.0], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision], t$95$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

      \[\leadsto \color{blue}{-1 \cdot \left({x}^{2} \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left({x}^{2} \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\left(x \cdot x\right) \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(x \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto x \cdot \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)\right)}\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{25} + \frac{253}{1000} \cdot \frac{1}{x}\right)\right)\right)}\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \left(\left(\mathsf{neg}\left(\frac{3}{25}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{253}{1000} \cdot \frac{1}{x}\right)\right)}\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{-3}{25} + \left(\mathsf{neg}\left(\color{blue}{\frac{253}{1000} \cdot \frac{1}{x}}\right)\right)\right)\right)\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{-3}{25} \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{253}{1000} \cdot \frac{1}{x}\right)\right) \cdot x}\right)\right) \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{-3}{25} \cdot x + \left(\left(\mathsf{neg}\left(\frac{253}{1000}\right)\right) \cdot \frac{1}{x}\right) \cdot x\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{-3}{25} \cdot x + \left(\frac{-253}{1000} \cdot \frac{1}{x}\right) \cdot x\right)\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{-3}{25} \cdot x + \frac{-253}{1000} \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right)\right) \]

      13. lft-mult-inverse N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{-3}{25} \cdot x + \frac{-253}{1000} \cdot 1\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{-3}{25} \cdot x + \frac{-253}{1000}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{-3}{25} \cdot x + \left(\mathsf{neg}\left(\frac{253}{1000}\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{-3}{25} \cdot x + \frac{-253}{1000}\right)\right) \]

      17. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{-253}{1000} + \color{blue}{\frac{-3}{25} \cdot x}\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-253}{1000}, \color{blue}{\left(\frac{-3}{25} \cdot x\right)}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-253}{1000}, \left(x \cdot \color{blue}{\frac{-3}{25}}\right)\right)\right) \]

      20. *-lowering-*.f64 98.8%

          \[\leadsto \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{-253}{1000}, \mathsf{*.f64}\left(x, \color{blue}{\frac{-3}{25}}\right)\right)\right) \]

   5. Simplified 98.8%

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

   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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{253}{1000} \cdot x\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\frac{253}{1000}}\right)\right) \]

      2. *-lowering-*.f64 98.5%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{253}{1000}}\right)\right) \]

   5. Simplified 98.5%

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

Alternative 3: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot -0.12\right)\\ \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x -0.12))))
   (if (<= x -4.1) t_0 (if (<= x 2.0) (- 1.0 (* x 0.253)) t_0))))

C

double code(double x) {
	double t_0 = x * (x * -0.12);
	double tmp;
	if (x <= -4.1) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} 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 <= (-4.1d0)) then
        tmp = t_0
    else if (x <= 2.0d0) then
        tmp = 1.0d0 - (x * 0.253d0)
    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 <= -4.1) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x * (x * -0.12)
	tmp = 0
	if x <= -4.1:
		tmp = t_0
	elif x <= 2.0:
		tmp = 1.0 - (x * 0.253)
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(x * Float64(x * -0.12))
	tmp = 0.0
	if (x <= -4.1)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = Float64(1.0 - Float64(x * 0.253));
	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 <= -4.1)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = 1.0 - (x * 0.253);
	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, -4.1], t$95$0, If[LessEqual[x, 2.0], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision], t$95$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

      \[\leadsto \color{blue}{\frac{-3}{25} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{-3}{25} \cdot \left(x \cdot \color{blue}{x}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-3}{25} \cdot x\right) \cdot \color{blue}{x} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-3}{25} \cdot x\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-3}{25} \cdot x\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-3}{25}}\right)\right) \]

      6. *-lowering-*.f64 97.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-3}{25}}\right)\right) \]

   5. Simplified 97.9%

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

   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

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{253}{1000} \cdot x\right)}\right) \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \color{blue}{\frac{253}{1000}}\right)\right) \]

      2. *-lowering-*.f64 98.5%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\frac{253}{1000}}\right)\right) \]

   5. Simplified 98.5%

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

Alternative 4: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot -0.12\right)\\ \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x -0.12)))) (if (<= x -4.1) 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 <= -4.1) {
		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 <= (-4.1d0)) 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 <= -4.1) {
		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 <= -4.1:
		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 <= -4.1)
		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 <= -4.1)
		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, -4.1], t$95$0, If[LessEqual[x, 2.0], 1.0, t$95$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

      \[\leadsto \color{blue}{\frac{-3}{25} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{-3}{25} \cdot \left(x \cdot \color{blue}{x}\right) \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{-3}{25} \cdot x\right) \cdot \color{blue}{x} \]

      3. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{-3}{25} \cdot x\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{-3}{25} \cdot x\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(x \cdot \color{blue}{\frac{-3}{25}}\right)\right) \]

      6. *-lowering-*.f64 97.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \color{blue}{\frac{-3}{25}}\right)\right) \]

   5. Simplified 97.9%

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

   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

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

   5. Simplified 97.1%

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

Alternative 5: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.12 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* -0.12 (* x x)))) (if (<= x -4.1) t_0 (if (<= x 2.0) 1.0 t_0))))

C

double code(double x) {
	double t_0 = -0.12 * (x * x);
	double tmp;
	if (x <= -4.1) {
		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 = (-0.12d0) * (x * x)
    if (x <= (-4.1d0)) 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 = -0.12 * (x * x);
	double tmp;
	if (x <= -4.1) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = -0.12 * (x * x)
	tmp = 0
	if x <= -4.1:
		tmp = t_0
	elif x <= 2.0:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x)
	t_0 = Float64(-0.12 * Float64(x * x))
	tmp = 0.0
	if (x <= -4.1)
		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 = -0.12 * (x * x);
	tmp = 0.0;
	if (x <= -4.1)
		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[(-0.12 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.1], t$95$0, If[LessEqual[x, 2.0], 1.0, t$95$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. Step-by-step derivation

      1. flip3-- N/A

         \[\leadsto \frac{{1}^{3} - {\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)}^{3}}{\color{blue}{1 \cdot 1 + \left(\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right) \cdot \left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right) + 1 \cdot \left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)}} \]

      2. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot 1 + \left(\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right) \cdot \left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right) + 1 \cdot \left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)}{{1}^{3} - {\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)}^{3}}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1 \cdot 1 + \left(\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right) \cdot \left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right) + 1 \cdot \left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)}{{1}^{3} - {\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)}^{3}}\right)}\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{\color{blue}{\frac{{1}^{3} - {\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)}^{3}}{1 \cdot 1 + \left(\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right) \cdot \left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right) + 1 \cdot \left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)}}}\right)\right) \]

      5. flip3-- N/A

         \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{1}{1 - \color{blue}{x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \color{blue}{\left(1 - x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(1 + \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \color{blue}{\left(\mathsf{neg}\left(x \cdot \left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)}\right)\right)\right) \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)\right)\right)}\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(\left(x \cdot \frac{3}{25} + \frac{253}{1000}\right)\right)\right)\right)\right)\right)\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(\left(\mathsf{neg}\left(x \cdot \frac{3}{25}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{253}{1000}\right)\right)}\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{-3}{25} \cdot {x}^{2}} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-3}{25}, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-3}{25}, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 97.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{-3}{25}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   7. Simplified 97.8%

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

   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

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

   5. Simplified 97.1%

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

   1. flip-+ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \frac{\frac{253}{1000} \cdot \frac{253}{1000} - \left(x \cdot \frac{3}{25}\right) \cdot \left(x \cdot \frac{3}{25}\right)}{\color{blue}{\frac{253}{1000} - x \cdot \frac{3}{25}}}\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(x \cdot \frac{1}{\color{blue}{\frac{\frac{253}{1000} - x \cdot \frac{3}{25}}{\frac{253}{1000} \cdot \frac{253}{1000} - \left(x \cdot \frac{3}{25}\right) \cdot \left(x \cdot \frac{3}{25}\right)}}}\right)\right) \]

   3. un-div-inv N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(\frac{x}{\color{blue}{\frac{\frac{253}{1000} - x \cdot \frac{3}{25}}{\frac{253}{1000} \cdot \frac{253}{1000} - \left(x \cdot \frac{3}{25}\right) \cdot \left(x \cdot \frac{3}{25}\right)}}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{253}{1000} - x \cdot \frac{3}{25}}{\frac{253}{1000} \cdot \frac{253}{1000} - \left(x \cdot \frac{3}{25}\right) \cdot \left(x \cdot \frac{3}{25}\right)}\right)}\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{\frac{253}{1000} \cdot \frac{253}{1000} - \left(x \cdot \frac{3}{25}\right) \cdot \left(x \cdot \frac{3}{25}\right)}{\frac{253}{1000} - x \cdot \frac{3}{25}}}}\right)\right)\right) \]

   6. flip-+ N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \left(\frac{1}{\frac{253}{1000} + \color{blue}{x \cdot \frac{3}{25}}}\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{253}{1000} + x \cdot \frac{3}{25}\right)}\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{253}{1000}, \color{blue}{\left(x \cdot \frac{3}{25}\right)}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\frac{253}{1000}, \mathsf{*.f64}\left(x, \color{blue}{\frac{3}{25}}\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{\frac{25}{3}}{x}\right)}\right)\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 97.5%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\frac{25}{3}, \color{blue}{x}\right)\right)\right) \]

7. Simplified 97.5%

   \[\leadsto 1 - \frac{x}{\color{blue}{\frac{8.333333333333334}{x}}} \]
8. Add Preprocessing

Alternative 7: 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

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

5. Simplified 47.5%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))