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

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 12

Speedup: 1.2×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * 0.04481 + N[(0.99229 * x + 1.0), $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. 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. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. associate-+l+ N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

6. Applied rewrites 100.0%

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

Alternative 2: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -40000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right) \cdot x - 2.0191289437, x, 2.30753\right) - x\\ \end{array} \end{array} \]

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 -40000.0) (not (<= t_0 4.0)))
     (- (/ 6.039053782637804 x) x)
     (-
      (fma
       (- (* (fma -1.7950336306565942 x 1.900161040244073) x) 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 <= -40000.0) || !(t_0 <= 4.0)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = fma(((fma(-1.7950336306565942, x, 1.900161040244073) * x) - 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 <= -40000.0) || !(t_0 <= 4.0))
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	else
		tmp = Float64(fma(Float64(Float64(fma(-1.7950336306565942, x, 1.900161040244073) * x) - 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, -40000.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(N[(-1.7950336306565942 * x + 1.900161040244073), $MachinePrecision] * x), $MachinePrecision] - 2.0191289437), $MachinePrecision] * 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) < -4e4 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

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481}}{x}} - x \]
   4. Step-by-step derivation

      1. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -4e4 < (-.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 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

      \[\leadsto \color{blue}{\left(\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right)\right)} - x \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right)} - x \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}\right) \cdot x} + \frac{230753}{100000}\right) - x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}, x, \frac{230753}{100000}\right)} - x \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{20191289437}{10000000000}}, x, \frac{230753}{100000}\right) - x \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x} - \frac{20191289437}{10000000000}, x, \frac{230753}{100000}\right) - x \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x} - \frac{20191289437}{10000000000}, x, \frac{230753}{100000}\right) - x \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}\right)} \cdot x - \frac{20191289437}{10000000000}, x, \frac{230753}{100000}\right) - x \]

      8. lower-fma.f64 99.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right)} \cdot x - 2.0191289437, x, 2.30753\right) - x \]

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right) \cdot x - 2.0191289437, x, 2.30753\right)} - x \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -40000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right) \cdot x - 2.0191289437, x, 2.30753\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -40000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right) \cdot x - 3.0191289437, x, 2.30753\right)\\ \end{array} \end{array} \]

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 -40000.0) (not (<= t_0 4.0)))
     (- (/ 6.039053782637804 x) x)
     (fma
      (- (* (fma -1.7950336306565942 x 1.900161040244073) x) 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 <= -40000.0) || !(t_0 <= 4.0)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = fma(((fma(-1.7950336306565942, x, 1.900161040244073) * x) - 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 <= -40000.0) || !(t_0 <= 4.0))
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	else
		tmp = fma(Float64(Float64(fma(-1.7950336306565942, x, 1.900161040244073) * x) - 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, -40000.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(N[(-1.7950336306565942 * x + 1.900161040244073), $MachinePrecision] * x), $MachinePrecision] - 3.0191289437), $MachinePrecision] * 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) < -4e4 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

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481}}{x}} - x \]
   4. Step-by-step derivation

      1. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -4e4 < (-.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 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

      \[\leadsto \color{blue}{\frac{230753}{100000} + x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) + \frac{230753}{100000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}\right) \cdot x} + \frac{230753}{100000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}, x, \frac{230753}{100000}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) - \frac{30191289437}{10000000000}}, x, \frac{230753}{100000}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x} - \frac{30191289437}{10000000000}, x, \frac{230753}{100000}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1900161040244073}{1000000000000000} + \frac{-179503363065659419717}{100000000000000000000} \cdot x\right) \cdot x} - \frac{30191289437}{10000000000}, x, \frac{230753}{100000}\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-179503363065659419717}{100000000000000000000} \cdot x + \frac{1900161040244073}{1000000000000000}\right)} \cdot x - \frac{30191289437}{10000000000}, x, \frac{230753}{100000}\right) \]

      8. lower-fma.f64 99.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right)} \cdot x - 3.0191289437, x, 2.30753\right) \]

   5. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right) \cdot x - 3.0191289437, x, 2.30753\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -40000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1.7950336306565942, x, 1.900161040244073\right) \cdot x - 3.0191289437, x, 2.30753\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -40000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 1.900161040244073, \mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\right)\\ \end{array} \end{array} \]

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 -40000.0) (not (<= t_0 4.0)))
     (- (/ 6.039053782637804 x) x)
     (fma (* x x) 1.900161040244073 (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 <= -40000.0) || !(t_0 <= 4.0)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = fma((x * x), 1.900161040244073, 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 <= -40000.0) || !(t_0 <= 4.0))
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	else
		tmp = fma(Float64(x * x), 1.900161040244073, 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, -40000.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * 1.900161040244073 + N[(-3.0191289437 * x + 2.30753), $MachinePrecision]), $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) < -4e4 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

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481}}{x}} - x \]
   4. Step-by-step derivation

      1. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -4e4 < (-.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 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. 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. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right) + \frac{230753}{100000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right) \cdot x} + \frac{230753}{100000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}, x, \frac{230753}{100000}\right)} \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\left(\mathsf{neg}\left(\frac{-30191289437}{10000000000}\right)\right)}, x, \frac{230753}{100000}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\color{blue}{\frac{-30191289437}{10000000000} \cdot 1}\right)\right), x, \frac{230753}{100000}\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\frac{-30191289437}{10000000000} \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right)\right), x, \frac{230753}{100000}\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-30191289437}{10000000000} \cdot \frac{1}{x}\right) \cdot x}\right)\right), x, \frac{230753}{100000}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right)} \cdot \frac{1}{x}\right) \cdot x\right)\right), x, \frac{230753}{100000}\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right) \cdot \frac{1}{x}\right)\right) \cdot x}, x, \frac{230753}{100000}\right) \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1900161040244073}{1000000000000000} \cdot x + \left(\left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right) \cdot \frac{1}{x}\right) \cdot x}, x, \frac{230753}{100000}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \left(\color{blue}{\frac{-30191289437}{10000000000}} \cdot \frac{1}{x}\right) \cdot x, x, \frac{230753}{100000}\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \color{blue}{\frac{-30191289437}{10000000000} \cdot \left(\frac{1}{x} \cdot x\right)}, x, \frac{230753}{100000}\right) \]

      13. lft-mult-inverse N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \frac{-30191289437}{10000000000} \cdot \color{blue}{1}, x, \frac{230753}{100000}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \color{blue}{\frac{-30191289437}{10000000000}}, x, \frac{230753}{100000}\right) \]

      15. lower-fma.f64 98.7

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right)}, x, 2.30753\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto {x}^{2} \cdot \color{blue}{\left(\left(\frac{1900161040244073}{1000000000000000} + \frac{\frac{230753}{100000}}{{x}^{2}}\right) - \frac{30191289437}{10000000000} \cdot \frac{1}{x}\right)} \]

   10. Applied rewrites 98.7%

       \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{1.900161040244073}, \mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -40000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, 1.900161040244073, \mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -40000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -2.0191289437\right), x, 2.30753\right) - x\\ \end{array} \end{array} \]

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 -40000.0) (not (<= t_0 4.0)))
     (- (/ 6.039053782637804 x) x)
     (- (fma (fma 1.900161040244073 x -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 <= -40000.0) || !(t_0 <= 4.0)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = fma(fma(1.900161040244073, x, -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 <= -40000.0) || !(t_0 <= 4.0))
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	else
		tmp = Float64(fma(fma(1.900161040244073, x, -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, -40000.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(N[(1.900161040244073 * x + -2.0191289437), $MachinePrecision] * 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) < -4e4 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

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481}}{x}} - x \]
   4. Step-by-step derivation

      1. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -4e4 < (-.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 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. 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. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right)\right)} - x \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right) + \frac{230753}{100000}\right)} - x \]

      2. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}\right) \cdot x} + \frac{230753}{100000}\right) - x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{20191289437}{10000000000}, x, \frac{230753}{100000}\right)} - x \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\left(\mathsf{neg}\left(\frac{-20191289437}{10000000000}\right)\right)}, x, \frac{230753}{100000}\right) - x \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\color{blue}{\frac{-20191289437}{10000000000} \cdot 1}\right)\right), x, \frac{230753}{100000}\right) - x \]

      6. lft-mult-inverse N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\frac{-20191289437}{10000000000} \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right)\right), x, \frac{230753}{100000}\right) - x \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-20191289437}{10000000000} \cdot \frac{1}{x}\right) \cdot x}\right)\right), x, \frac{230753}{100000}\right) - x \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right)} \cdot \frac{1}{x}\right) \cdot x\right)\right), x, \frac{230753}{100000}\right) - x \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right) \cdot \frac{1}{x}\right)\right) \cdot x}, x, \frac{230753}{100000}\right) - x \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1900161040244073}{1000000000000000} \cdot x + \left(\left(\mathsf{neg}\left(\frac{20191289437}{10000000000}\right)\right) \cdot \frac{1}{x}\right) \cdot x}, x, \frac{230753}{100000}\right) - x \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \left(\color{blue}{\frac{-20191289437}{10000000000}} \cdot \frac{1}{x}\right) \cdot x, x, \frac{230753}{100000}\right) - x \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \color{blue}{\frac{-20191289437}{10000000000} \cdot \left(\frac{1}{x} \cdot x\right)}, x, \frac{230753}{100000}\right) - x \]

      13. lft-mult-inverse N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \frac{-20191289437}{10000000000} \cdot \color{blue}{1}, x, \frac{230753}{100000}\right) - x \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \color{blue}{\frac{-20191289437}{10000000000}}, x, \frac{230753}{100000}\right) - x \]

      15. lower-fma.f64 98.7

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(1.900161040244073, x, -2.0191289437\right)}, x, 2.30753\right) - x \]

   9. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -2.0191289437\right), x, 2.30753\right)} - x \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -40000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -2.0191289437\right), x, 2.30753\right) - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -40000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)\\ \end{array} \end{array} \]

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 -40000.0) (not (<= t_0 4.0)))
     (- (/ 6.039053782637804 x) x)
     (fma (fma 1.900161040244073 x -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 <= -40000.0) || !(t_0 <= 4.0)) {
		tmp = (6.039053782637804 / x) - x;
	} else {
		tmp = fma(fma(1.900161040244073, x, -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 <= -40000.0) || !(t_0 <= 4.0))
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	else
		tmp = fma(fma(1.900161040244073, x, -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, -40000.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], N[(N[(1.900161040244073 * x + -3.0191289437), $MachinePrecision] * 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) < -4e4 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

      \[\leadsto \color{blue}{\frac{\frac{27061}{4481}}{x}} - x \]
   4. Step-by-step derivation

      1. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

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

   if -4e4 < (-.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 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. 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. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right) + \frac{230753}{100000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right) \cdot x} + \frac{230753}{100000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}, x, \frac{230753}{100000}\right)} \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\left(\mathsf{neg}\left(\frac{-30191289437}{10000000000}\right)\right)}, x, \frac{230753}{100000}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\color{blue}{\frac{-30191289437}{10000000000} \cdot 1}\right)\right), x, \frac{230753}{100000}\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\frac{-30191289437}{10000000000} \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right)\right), x, \frac{230753}{100000}\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-30191289437}{10000000000} \cdot \frac{1}{x}\right) \cdot x}\right)\right), x, \frac{230753}{100000}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right)} \cdot \frac{1}{x}\right) \cdot x\right)\right), x, \frac{230753}{100000}\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right) \cdot \frac{1}{x}\right)\right) \cdot x}, x, \frac{230753}{100000}\right) \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1900161040244073}{1000000000000000} \cdot x + \left(\left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right) \cdot \frac{1}{x}\right) \cdot x}, x, \frac{230753}{100000}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \left(\color{blue}{\frac{-30191289437}{10000000000}} \cdot \frac{1}{x}\right) \cdot x, x, \frac{230753}{100000}\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \color{blue}{\frac{-30191289437}{10000000000} \cdot \left(\frac{1}{x} \cdot x\right)}, x, \frac{230753}{100000}\right) \]

      13. lft-mult-inverse N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \frac{-30191289437}{10000000000} \cdot \color{blue}{1}, x, \frac{230753}{100000}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \color{blue}{\frac{-30191289437}{10000000000}}, x, \frac{230753}{100000}\right) \]

      15. lower-fma.f64 98.7

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right)}, x, 2.30753\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -40000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -40000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)\\ \end{array} \end{array} \]

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 -40000.0) (not (<= t_0 4.0)))
     (- x)
     (fma (fma 1.900161040244073 x -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 <= -40000.0) || !(t_0 <= 4.0)) {
		tmp = -x;
	} else {
		tmp = fma(fma(1.900161040244073, x, -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 <= -40000.0) || !(t_0 <= 4.0))
		tmp = Float64(-x);
	else
		tmp = fma(fma(1.900161040244073, x, -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, -40000.0], N[Not[LessEqual[t$95$0, 4.0]], $MachinePrecision]], (-x), N[(N[(1.900161040244073 * x + -3.0191289437), $MachinePrecision] * 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) < -4e4 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 99.0

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

   5. Applied rewrites 99.0%

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

   if -4e4 < (-.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 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. 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. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{230753}{100000} + x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right) + \frac{230753}{100000}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}\right) \cdot x} + \frac{230753}{100000} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \frac{30191289437}{10000000000}, x, \frac{230753}{100000}\right)} \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\left(\mathsf{neg}\left(\frac{-30191289437}{10000000000}\right)\right)}, x, \frac{230753}{100000}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\color{blue}{\frac{-30191289437}{10000000000} \cdot 1}\right)\right), x, \frac{230753}{100000}\right) \]

      6. lft-mult-inverse N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\frac{-30191289437}{10000000000} \cdot \color{blue}{\left(\frac{1}{x} \cdot x\right)}\right)\right), x, \frac{230753}{100000}\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{-30191289437}{10000000000} \cdot \frac{1}{x}\right) \cdot x}\right)\right), x, \frac{230753}{100000}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right)} \cdot \frac{1}{x}\right) \cdot x\right)\right), x, \frac{230753}{100000}\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x - \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right) \cdot \frac{1}{x}\right)\right) \cdot x}, x, \frac{230753}{100000}\right) \]

      10. fp-cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1900161040244073}{1000000000000000} \cdot x + \left(\left(\mathsf{neg}\left(\frac{30191289437}{10000000000}\right)\right) \cdot \frac{1}{x}\right) \cdot x}, x, \frac{230753}{100000}\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \left(\color{blue}{\frac{-30191289437}{10000000000}} \cdot \frac{1}{x}\right) \cdot x, x, \frac{230753}{100000}\right) \]

      12. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \color{blue}{\frac{-30191289437}{10000000000} \cdot \left(\frac{1}{x} \cdot x\right)}, x, \frac{230753}{100000}\right) \]

      13. lft-mult-inverse N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \frac{-30191289437}{10000000000} \cdot \color{blue}{1}, x, \frac{230753}{100000}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{1900161040244073}{1000000000000000} \cdot x + \color{blue}{\frac{-30191289437}{10000000000}}, x, \frac{230753}{100000}\right) \]

      15. lower-fma.f64 98.7

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right)}, x, 2.30753\right) \]

   7. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -40000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(1.900161040244073, x, -3.0191289437\right), x, 2.30753\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -40000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\\ \end{array} \end{array} \]

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 -40000.0) (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 <= -40000.0) || !(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 <= -40000.0) || !(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, -40000.0], 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) < -4e4 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 99.0

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

   5. Applied rewrites 99.0%

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

   if -4e4 < (-.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 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

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -40000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-3.0191289437, x, 2.30753\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x\\ \mathbf{if}\;t\_0 \leq -40000 \lor \neg \left(t\_0 \leq 4\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753\\ \end{array} \end{array} \]

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 -40000.0) (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 <= -40000.0) || !(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 <= (-40000.0d0)) .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 <= -40000.0) || !(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 <= -40000.0) 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 <= -40000.0) || !(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 <= -40000.0) || ~((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, -40000.0], 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) < -4e4 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 99.0

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

   5. Applied rewrites 99.0%

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

   if -4e4 < (-.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 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. 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. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

      8. associate-+l+ N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 97.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq -40000 \lor \neg \left(\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \leq 4\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 11: 98.7% 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 98.2

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

5. Applied rewrites 98.2%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{\color{blue}{\mathsf{fma}\left(0.99229, x, 1\right)}} - x \]
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 98.2

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

7. Applied rewrites 98.2%

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

Alternative 12: 50.6% 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

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

   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. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. associate-+l+ N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Taylor expanded in x around 0

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

9. Applied rewrites 47.2%

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

10. Final simplification 47.2%

    \[\leadsto 2.30753 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(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))