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

Percentage Accurate: 100.0% → 100.0%

Time: 9.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.27061 + 2.30753}{\left(1 + x \cdot 0.99229\right) + 0.04481 \cdot {x}^{2}} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  (/
   (+ (* x 0.27061) 2.30753)
   (+ (+ 1.0 (* x 0.99229)) (* 0.04481 (pow x 2.0))))
  x))

C

double code(double x) {
	return (((x * 0.27061) + 2.30753) / ((1.0 + (x * 0.99229)) + (0.04481 * pow(x, 2.0)))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((x * 0.27061d0) + 2.30753d0) / ((1.0d0 + (x * 0.99229d0)) + (0.04481d0 * (x ** 2.0d0)))) - x
end function

Java

public static double code(double x) {
	return (((x * 0.27061) + 2.30753) / ((1.0 + (x * 0.99229)) + (0.04481 * Math.pow(x, 2.0)))) - x;
}

Python

def code(x):
	return (((x * 0.27061) + 2.30753) / ((1.0 + (x * 0.99229)) + (0.04481 * math.pow(x, 2.0)))) - x

Julia

function code(x)
	return Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(Float64(1.0 + Float64(x * 0.99229)) + Float64(0.04481 * (x ^ 2.0)))) - x)
end

MATLAB

function tmp = code(x)
	tmp = (((x * 0.27061) + 2.30753) / ((1.0 + (x * 0.99229)) + (0.04481 * (x ^ 2.0)))) - x;
end

Wolfram

code[x_] := N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(N[(1.0 + N[(x * 0.99229), $MachinePrecision]), $MachinePrecision] + N[(0.04481 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. fma-define 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   3. +-commutative 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}} - x \]

   4. fma-define 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}} - x \]

   5. +-commutative 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.04481 + 0.99229}, 1\right)} - x \]

   6. fma-define 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)}, 1\right)} - x \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} - x} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{x \cdot \mathsf{fma}\left(x, 0.04481, 0.99229\right) + 1}} - x \]

   2. fma-undefine 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{x \cdot \color{blue}{\left(x \cdot 0.04481 + 0.99229\right)} + 1} - x \]

   3. +-commutative 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{x \cdot \color{blue}{\left(0.99229 + x \cdot 0.04481\right)} + 1} - x \]

   4. +-commutative 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)}} - x \]

   5. distribute-rgt-in 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{1 + \color{blue}{\left(0.99229 \cdot x + \left(x \cdot 0.04481\right) \cdot x\right)}} - x \]

   6. associate-+r+ 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\left(1 + 0.99229 \cdot x\right) + \left(x \cdot 0.04481\right) \cdot x}} - x \]

   7. *-commutative 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\left(1 + \color{blue}{x \cdot 0.99229}\right) + \left(x \cdot 0.04481\right) \cdot x} - x \]

   8. *-commutative 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\left(1 + x \cdot 0.99229\right) + \color{blue}{\left(0.04481 \cdot x\right)} \cdot x} - x \]

   9. associate-*l* 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\left(1 + x \cdot 0.99229\right) + \color{blue}{0.04481 \cdot \left(x \cdot x\right)}} - x \]

   10. pow2 100.0%

       \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\left(1 + x \cdot 0.99229\right) + 0.04481 \cdot \color{blue}{{x}^{2}}} - x \]

6. Applied egg-rr 100.0%

   \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\left(1 + x \cdot 0.99229\right) + 0.04481 \cdot {x}^{2}}} - x \]
7. Step-by-step derivation

   1. fma-undefine 100.0%

      \[\leadsto \frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{\left(1 + x \cdot 0.99229\right) + 0.04481 \cdot {x}^{2}} - x \]

8. Applied egg-rr 100.0%

   \[\leadsto \frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{\left(1 + x \cdot 0.99229\right) + 0.04481 \cdot {x}^{2}} - x \]
9. Add Preprocessing

Alternative 2: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;\left(2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 2.0191289437\right)\right) - x\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.0)
   (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x)
   (if (<= x 1.2)
     (- (+ 2.30753 (* x (- (* x 1.900161040244073) 2.0191289437))) x)
     (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -5.0) {
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / x) - x;
	} else if (x <= 1.2) {
		tmp = (2.30753 + (x * ((x * 1.900161040244073) - 2.0191289437))) - x;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.0d0)) then
        tmp = ((6.039053782637804d0 - (82.23527511657367d0 / x)) / x) - x
    else if (x <= 1.2d0) then
        tmp = (2.30753d0 + (x * ((x * 1.900161040244073d0) - 2.0191289437d0))) - x
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.0) {
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / x) - x;
	} else if (x <= 1.2) {
		tmp = (2.30753 + (x * ((x * 1.900161040244073) - 2.0191289437))) - x;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.0:
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / x) - x
	elif x <= 1.2:
		tmp = (2.30753 + (x * ((x * 1.900161040244073) - 2.0191289437))) - x
	else:
		tmp = -x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.0)
		tmp = Float64(Float64(Float64(6.039053782637804 - Float64(82.23527511657367 / x)) / x) - x);
	elseif (x <= 1.2)
		tmp = Float64(Float64(2.30753 + Float64(x * Float64(Float64(x * 1.900161040244073) - 2.0191289437))) - x);
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.0)
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / x) - x;
	elseif (x <= 1.2)
		tmp = (2.30753 + (x * ((x * 1.900161040244073) - 2.0191289437))) - x;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.0], N[(N[(N[(6.039053782637804 - N[(82.23527511657367 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 1.2], N[(N[(2.30753 + N[(x * N[(N[(x * 1.900161040244073), $MachinePrecision] - 2.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision], (-x)]]

Derivation

1. Split input into 3 regimes

2. if x < -5

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x \]
   4. Step-by-step derivation

      1. associate-*r/ 99.6%

         \[\leadsto \frac{6.039053782637804 - \color{blue}{\frac{82.23527511657367 \cdot 1}{x}}}{x} - x \]

      2. metadata-eval 99.6%

         \[\leadsto \frac{6.039053782637804 - \frac{\color{blue}{82.23527511657367}}{x}}{x} - x \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x}} - x \]

   if -5 < x < 1.19999999999999996

   1. Initial program 99.9%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.1%

      \[\leadsto \color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right)} - x \]

   if 1.19999999999999996 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{2.30753} - x \]

   4. Taylor expanded in x around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;\left(2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 2.0191289437\right)\right) - x\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.27061 + 2.30753}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ (* x 0.27061) 2.30753) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return (((x * 0.27061) + 2.30753) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((x * 0.27061d0) + 2.30753d0) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return (((x * 0.27061) + 2.30753) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return (((x * 0.27061) + 2.30753) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = (((x * 0.27061) + 2.30753) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto \frac{x \cdot 0.27061 + 2.30753}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
4. Add Preprocessing

Alternative 4: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.4:\\ \;\;\;\;\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x} - x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;2.30753 + x \cdot -3.0191289437\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5.4)
   (- (/ (- 6.039053782637804 (/ 82.23527511657367 x)) x) x)
   (if (<= x 1.15) (+ 2.30753 (* x -3.0191289437)) (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -5.4) {
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / x) - x;
	} else if (x <= 1.15) {
		tmp = 2.30753 + (x * -3.0191289437);
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-5.4d0)) then
        tmp = ((6.039053782637804d0 - (82.23527511657367d0 / x)) / x) - x
    else if (x <= 1.15d0) then
        tmp = 2.30753d0 + (x * (-3.0191289437d0))
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -5.4) {
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / x) - x;
	} else if (x <= 1.15) {
		tmp = 2.30753 + (x * -3.0191289437);
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -5.4:
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / x) - x
	elif x <= 1.15:
		tmp = 2.30753 + (x * -3.0191289437)
	else:
		tmp = -x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -5.4)
		tmp = Float64(Float64(Float64(6.039053782637804 - Float64(82.23527511657367 / x)) / x) - x);
	elseif (x <= 1.15)
		tmp = Float64(2.30753 + Float64(x * -3.0191289437));
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -5.4)
		tmp = ((6.039053782637804 - (82.23527511657367 / x)) / x) - x;
	elseif (x <= 1.15)
		tmp = 2.30753 + (x * -3.0191289437);
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -5.4], N[(N[(N[(6.039053782637804 - N[(82.23527511657367 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 1.15], N[(2.30753 + N[(x * -3.0191289437), $MachinePrecision]), $MachinePrecision], (-x)]]

Derivation

1. Split input into 3 regimes

2. if x < -5.4000000000000004

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.6%

      \[\leadsto \color{blue}{\frac{6.039053782637804 - 82.23527511657367 \cdot \frac{1}{x}}{x}} - x \]
   4. Step-by-step derivation

      1. associate-*r/ 99.6%

         \[\leadsto \frac{6.039053782637804 - \color{blue}{\frac{82.23527511657367 \cdot 1}{x}}}{x} - x \]

      2. metadata-eval 99.6%

         \[\leadsto \frac{6.039053782637804 - \frac{\color{blue}{82.23527511657367}}{x}}{x} - x \]

   5. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{6.039053782637804 - \frac{82.23527511657367}{x}}{x}} - x \]

   if -5.4000000000000004 < x < 1.1499999999999999

   1. Initial program 99.9%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.1%

      \[\leadsto \color{blue}{\left(2.30753 + x \cdot \left(1.900161040244073 \cdot x - 2.0191289437\right)\right)} - x \]

   4. Taylor expanded in x around 0 97.6%

      \[\leadsto \left(2.30753 + x \cdot \color{blue}{-2.0191289437}\right) - x \]

   5. Taylor expanded in x around 0 97.6%

      \[\leadsto \color{blue}{2.30753 + -3.0191289437 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 97.6%

         \[\leadsto 2.30753 + \color{blue}{x \cdot -3.0191289437} \]

   7. Simplified 97.6%

      \[\leadsto \color{blue}{2.30753 + x \cdot -3.0191289437} \]

   if 1.1499999999999999 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 99.8%

      \[\leadsto \color{blue}{2.30753} - x \]

   4. Taylor expanded in x around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   6. Simplified 100.0%

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

Alternative 5: 98.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.27061 + 2.30753}{1 + x \cdot 0.99229} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ (* x 0.27061) 2.30753) (+ 1.0 (* x 0.99229))) x))

C

double code(double x) {
	return (((x * 0.27061) + 2.30753) / (1.0 + (x * 0.99229))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((x * 0.27061d0) + 2.30753d0) / (1.0d0 + (x * 0.99229d0))) - x
end function

Java

public static double code(double x) {
	return (((x * 0.27061) + 2.30753) / (1.0 + (x * 0.99229))) - x;
}

Python

def code(x):
	return (((x * 0.27061) + 2.30753) / (1.0 + (x * 0.99229))) - x

Julia

function code(x)
	return Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(1.0 + Float64(x * 0.99229))) - x)
end

MATLAB

function tmp = code(x)
	tmp = (((x * 0.27061) + 2.30753) / (1.0 + (x * 0.99229))) - x;
end

Wolfram

code[x_] := N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(1.0 + N[(x * 0.99229), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.1%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} - x \]
4. Step-by-step derivation

   1. *-commutative 98.1%

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} - x \]

5. Simplified 98.1%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} - x \]

6. Final simplification 98.1%

   \[\leadsto \frac{x \cdot 0.27061 + 2.30753}{1 + x \cdot 0.99229} - x \]
7. Add Preprocessing

Alternative 6: 98.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ 2.30753 - x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 2.30753 x))

C

double code(double x) {
	return 2.30753 - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.30753d0 - x
end function

Java

public static double code(double x) {
	return 2.30753 - x;
}

Python

def code(x):
	return 2.30753 - x

Julia

function code(x)
	return Float64(2.30753 - x)
end

MATLAB

function tmp = code(x)
	tmp = 2.30753 - x;
end

Wolfram

code[x_] := N[(2.30753 - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 97.0%

   \[\leadsto \color{blue}{2.30753} - x \]
4. Add Preprocessing

Alternative 7: 51.5% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- x))

C

double code(double x) {
	return -x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -x
end function

Java

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

Python

def code(x):
	return -x

Julia

function code(x)
	return Float64(-x)
end

MATLAB

function tmp = code(x)
	tmp = -x;
end

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 97.0%

   \[\leadsto \color{blue}{2.30753} - x \]

4. Taylor expanded in x around inf 50.9%

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

   1. neg-mul-1 50.9%

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

6. Simplified 50.9%

   \[\leadsto \color{blue}{-x} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))