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

Percentage Accurate: 100.0% → 100.0%

Time: 6.4s

Alternatives: 8

Speedup: N/A×

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.04481, x, 0.99229\right), x, 1\right)\\ \mathsf{fma}\left(x, \frac{0.27061}{t\_0}, \frac{2.30753}{t\_0} - x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fma 0.04481 x 0.99229) x 1.0)))
   (fma x (/ 0.27061 t_0) (- (/ 2.30753 t_0) x))))

C

double code(double x) {
	double t_0 = fma(fma(0.04481, x, 0.99229), x, 1.0);
	return fma(x, (0.27061 / t_0), ((2.30753 / t_0) - x));
}

Julia

function code(x)
	t_0 = fma(fma(0.04481, x, 0.99229), x, 1.0)
	return fma(x, Float64(0.27061 / t_0), Float64(Float64(2.30753 / t_0) - x))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(0.04481 * x + 0.99229), $MachinePrecision] * x + 1.0), $MachinePrecision]}, N[(x * N[(0.27061 / t$95$0), $MachinePrecision] + N[(N[(2.30753 / t$95$0), $MachinePrecision] - x), $MachinePrecision]), $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. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   4. +-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   5. div-add N/A

      \[\leadsto \color{blue}{\left(\frac{x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} - x \]

   6. associate--l+ N/A

      \[\leadsto \color{blue}{\frac{x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \]

   8. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{\frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} + \left(\frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{\frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}, \frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

4. Applied rewrites 100.0%

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

Alternative 2: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq 1 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2.0191289437, x, 2.30753\right) - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (or (<= t_0 1.0) (not (<= t_0 4.0)))
     (- x)
     (- (fma -2.0191289437 x 2.30753) x))))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if ((t_0 <= 1.0) || !(t_0 <= 4.0)) {
		tmp = -x;
	} else {
		tmp = fma(-2.0191289437, x, 2.30753) - x;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if ((t_0 <= 1.0) || !(t_0 <= 4.0))
		tmp = Float64(-x);
	else
		tmp = Float64(fma(-2.0191289437, x, 2.30753) - x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, 1.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], (-x), N[(N[(-2.0191289437 * x + 2.30753), $MachinePrecision] - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 1 or 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) 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 inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 1 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   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

      \[\leadsto \color{blue}{\left(\frac{230753}{100000} + \frac{-20191289437}{10000000000} \cdot x\right)} - x \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-20191289437}{10000000000} \cdot x + \frac{230753}{100000}\right)} - x \]

      2. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(-2.0191289437, x, 2.30753\right)} - x \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2.0191289437, x, 2.30753\right)} - x \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 1 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2.0191289437, x, 2.30753\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq 1 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (or (<= t_0 1.0) (not (<= t_0 4.0)))
     (- x)
     (fma -3.0191289437 x 2.30753))))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if ((t_0 <= 1.0) || !(t_0 <= 4.0)) {
		tmp = -x;
	} else {
		tmp = fma(-3.0191289437, x, 2.30753);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if ((t_0 <= 1.0) || !(t_0 <= 4.0))
		tmp = Float64(-x);
	else
		tmp = fma(-3.0191289437, x, 2.30753);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, 1.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], (-x), N[(-3.0191289437 * x + 2.30753), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 1 or 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) 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 inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 1 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   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

      \[\leadsto \color{blue}{\frac{230753}{100000} + \frac{-30191289437}{10000000000} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-30191289437}{10000000000} \cdot x + \frac{230753}{100000}} \]

      2. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(-3.0191289437, x, 2.30753\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-3.0191289437, x, 2.30753\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 1 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -2 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0
         (-
          (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481)))))
          x)))
   (if (or (<= t_0 -2.0) (not (<= t_0 4.0))) (- x) 2.30753)))

C

double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if ((t_0 <= -2.0) || !(t_0 <= 4.0)) {
		tmp = -x;
	} else {
		tmp = 2.30753;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
    if ((t_0 <= (-2.0d0)) .or. (.not. (t_0 <= 4.0d0))) then
        tmp = -x
    else
        tmp = 2.30753d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	double tmp;
	if ((t_0 <= -2.0) || !(t_0 <= 4.0)) {
		tmp = -x;
	} else {
		tmp = 2.30753;
	}
	return tmp;
}

Python

def code(x):
	t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x
	tmp = 0
	if (t_0 <= -2.0) or not (t_0 <= 4.0):
		tmp = -x
	else:
		tmp = 2.30753
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
	tmp = 0.0
	if ((t_0 <= -2.0) || !(t_0 <= 4.0))
		tmp = Float64(-x);
	else
		tmp = 2.30753;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
	tmp = 0.0;
	if ((t_0 <= -2.0) || ~((t_0 <= 4.0)))
		tmp = -x;
	else
		tmp = 2.30753;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, -2.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], (-x), 2.30753]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < -2 or 4 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) 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 inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. lower-neg.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -2 < (-.f64 (/.f64 (+.f64 #s(literal 230753/100000 binary64) (*.f64 x #s(literal 27061/100000 binary64))) (+.f64 #s(literal 1 binary64) (*.f64 x (+.f64 #s(literal 99229/100000 binary64) (*.f64 x #s(literal 4481/100000 binary64)))))) x) < 4

   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. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      4. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

      5. div-add N/A

         \[\leadsto \color{blue}{\left(\frac{x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} - x \]

      6. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{\frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} + \left(\frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{\frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}, \frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000}} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.6%

      \[\leadsto \color{blue}{2.30753} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -2 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

   5. lower-fma.f64 100.0

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

   6. lift-+.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

   7. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

   9. *-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

   10. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 6: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(fma(0.27061, x, 2.30753) / fma(fma(0.04481, x, 0.99229), x, 1.0)) - x)
end

Wolfram

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

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   4. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{27061}{100000} \cdot x} + \frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   5. lower-fma.f64 100.0

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

   6. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) + 1}} - x \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + 1} - x \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\color{blue}{\left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right) \cdot x} + 1} - x \]

   10. lower-fma.f64 100.0

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

   11. lift-+.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{99229}{100000} + x \cdot \frac{4481}{100000}}, x, 1\right)} - x \]

   12. +-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000} + \frac{99229}{100000}}, x, 1\right)} - x \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{x \cdot \frac{4481}{100000}} + \frac{99229}{100000}, x, 1\right)} - x \]

   14. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{27061}{100000}, x, \frac{230753}{100000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{4481}{100000} \cdot x} + \frac{99229}{100000}, x, 1\right)} - x \]

   15. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 7: 98.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(0.99229, x, 1\right)} - x \end{array} \]

FPCore

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

C

double code(double x) {
	return (fma(x, 0.27061, 2.30753) / fma(0.99229, x, 1.0)) - x;
}

Julia

function code(x)
	return Float64(Float64(fma(x, 0.27061, 2.30753) / fma(0.99229, x, 1.0)) - x)
end

Wolfram

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

   \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{1 + \frac{99229}{100000} \cdot x}} - x \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{\color{blue}{\frac{99229}{100000} \cdot x + 1}} - x \]

   2. lower-fma.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{\mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)} - x \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{\mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)} - x \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}} + \frac{230753}{100000}}{\mathsf{fma}\left(\frac{99229}{100000}, x, 1\right)} - x \]

   4. lower-fma.f64 99.2

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

7. Applied rewrites 99.2%

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

Alternative 8: 51.4% accurate, 39.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 2.30753)

C

double code(double x) {
	return 2.30753;
}

Fortran

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

Java

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

Python

def code(x):
	return 2.30753

Julia

function code(x)
	return 2.30753
end

MATLAB

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

Wolfram

code[x_] := 2.30753

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. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x} \]

   2. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} - x \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{230753}{100000} + x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   4. +-commutative N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000} + \frac{230753}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x \]

   5. div-add N/A

      \[\leadsto \color{blue}{\left(\frac{x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}\right)} - x \]

   6. associate--l+ N/A

      \[\leadsto \color{blue}{\frac{x \cdot \frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \frac{27061}{100000}}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} + \left(\frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \]

   8. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{\frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}} + \left(\frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{\frac{27061}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)}, \frac{\frac{230753}{100000}}{1 + x \cdot \left(\frac{99229}{100000} + x \cdot \frac{4481}{100000}\right)} - x\right)} \]

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{230753}{100000}} \]
6. Step-by-step derivation

7. Applied rewrites 54.5%

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

8. Final simplification 54.5%

   \[\leadsto 2.30753 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024320 
(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))