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

Percentage Accurate: 58.0% → 96.4%

Time: 14.8s

Alternatives: 18

Speedup: 4.4×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/
  (*
   (- x 2.0)
   (+
    (*
     (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y)
     x)
    z))
  (+
   (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x)
   47.066876606)))

C

double code(double x, double y, double z) {
	return ((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - 2.0d0) * ((((((((x * 4.16438922228d0) + 78.6994924154d0) * x) + 137.519416416d0) * x) + y) * x) + z)) / (((((((x + 43.3400022514d0) * x) + 263.505074721d0) * x) + 313.399215894d0) * x) + 47.066876606d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606);
}

Python

def code(x, y, z):
	return ((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - 2.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision] * x), $MachinePrecision] + 137.519416416), $MachinePrecision] * x), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(x + 43.3400022514), $MachinePrecision] * x), $MachinePrecision] + 263.505074721), $MachinePrecision] * x), $MachinePrecision] + 313.399215894), $MachinePrecision] * x), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]

Initial Program: 58.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/
  (*
   (- x 2.0)
   (+
    (*
     (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y)
     x)
    z))
  (+
   (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x)
   47.066876606)))

C

double code(double x, double y, double z) {
	return ((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x - 2.0d0) * ((((((((x * 4.16438922228d0) + 78.6994924154d0) * x) + 137.519416416d0) * x) + y) * x) + z)) / (((((((x + 43.3400022514d0) * x) + 263.505074721d0) * x) + 313.399215894d0) * x) + 47.066876606d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606);
}

Python

def code(x, y, z):
	return ((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - 2.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision] * x), $MachinePrecision] + 137.519416416), $MachinePrecision] * x), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(x + 43.3400022514), $MachinePrecision] * x), $MachinePrecision] + 263.505074721), $MachinePrecision] * x), $MachinePrecision] + 313.399215894), $MachinePrecision] * x), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x\\ \mathbf{if}\;\frac{\left(z + \left(y + \left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x - 2\right)}{t\_0} \leq 10^{+290}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{\mathsf{fma}\left(17.342137594641823, x \cdot x, -6193.6101064416025\right)}{\mathsf{fma}\left(4.16438922228, x, -78.6994924154\right)} \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right) \cdot \left(x - 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          47.066876606
          (*
           (+ 313.399215894 (* (+ 263.505074721 (* (+ 43.3400022514 x) x)) x))
           x))))
   (if (<=
        (/
         (*
          (+
           z
           (*
            (+
             y
             (*
              (+ 137.519416416 (* (+ 78.6994924154 (* 4.16438922228 x)) x))
              x))
            x))
          (- x 2.0))
         t_0)
        1e+290)
     (/
      (*
       (+
        (*
         (+
          (*
           (+
            (*
             (/
              (fma 17.342137594641823 (* x x) -6193.6101064416025)
              (fma 4.16438922228 x -78.6994924154))
             x)
            137.519416416)
           x)
          y)
         x)
        z)
       (- x 2.0))
      t_0)
     (*
      (-
       (/
        (-
         -110.1139242984811
         (/ (- (+ (/ 130977.50649958357 x) -3655.1204654076414) (/ y x)) x))
        x)
       -4.16438922228)
      x))))

C

double code(double x, double y, double z) {
	double t_0 = 47.066876606 + ((313.399215894 + ((263.505074721 + ((43.3400022514 + x) * x)) * x)) * x);
	double tmp;
	if ((((z + ((y + ((137.519416416 + ((78.6994924154 + (4.16438922228 * x)) * x)) * x)) * x)) * (x - 2.0)) / t_0) <= 1e+290) {
		tmp = ((((((((fma(17.342137594641823, (x * x), -6193.6101064416025) / fma(4.16438922228, x, -78.6994924154)) * x) + 137.519416416) * x) + y) * x) + z) * (x - 2.0)) / t_0;
	} else {
		tmp = (((-110.1139242984811 - ((((130977.50649958357 / x) + -3655.1204654076414) - (y / x)) / x)) / x) - -4.16438922228) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(47.066876606 + Float64(Float64(313.399215894 + Float64(Float64(263.505074721 + Float64(Float64(43.3400022514 + x) * x)) * x)) * x))
	tmp = 0.0
	if (Float64(Float64(Float64(z + Float64(Float64(y + Float64(Float64(137.519416416 + Float64(Float64(78.6994924154 + Float64(4.16438922228 * x)) * x)) * x)) * x)) * Float64(x - 2.0)) / t_0) <= 1e+290)
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(fma(17.342137594641823, Float64(x * x), -6193.6101064416025) / fma(4.16438922228, x, -78.6994924154)) * x) + 137.519416416) * x) + y) * x) + z) * Float64(x - 2.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(-110.1139242984811 - Float64(Float64(Float64(Float64(130977.50649958357 / x) + -3655.1204654076414) - Float64(y / x)) / x)) / x) - -4.16438922228) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(47.066876606 + N[(N[(313.399215894 + N[(N[(263.505074721 + N[(N[(43.3400022514 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(z + N[(N[(y + N[(N[(137.519416416 + N[(N[(78.6994924154 + N[(4.16438922228 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], 1e+290], N[(N[(N[(N[(N[(N[(N[(N[(N[(N[(17.342137594641823 * N[(x * x), $MachinePrecision] + -6193.6101064416025), $MachinePrecision] / N[(4.16438922228 * x + -78.6994924154), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + 137.519416416), $MachinePrecision] * x), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision] + z), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(-110.1139242984811 - N[(N[(N[(N[(130977.50649958357 / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64))) < 1.00000000000000006e290

   1. Initial program 99.0%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\color{blue}{\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right)} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      2. flip-+ N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\color{blue}{\frac{\left(x \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) - \frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\color{blue}{\frac{\left(x \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) - \frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      4. sub-neg N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\color{blue}{\left(x \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\color{blue}{\left(x \cdot \frac{104109730557}{25000000000}\right)} \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      6. *-commutative N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\color{blue}{\left(\frac{104109730557}{25000000000} \cdot x\right)} \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\left(\frac{104109730557}{25000000000} \cdot x\right) \cdot \color{blue}{\left(x \cdot \frac{104109730557}{25000000000}\right)} + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\left(\frac{104109730557}{25000000000} \cdot x\right) \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      9. swap-sqr N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot x\right)} + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{104109730557}{25000000000} \cdot \frac{104109730557}{25000000000}, x \cdot x, \mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\mathsf{fma}\left(\color{blue}{\frac{10838835996651139530249}{625000000000000000000}}, x \cdot x, \mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\mathsf{fma}\left(\frac{10838835996651139530249}{625000000000000000000}, \color{blue}{x \cdot x}, \mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\mathsf{fma}\left(\frac{10838835996651139530249}{625000000000000000000}, x \cdot x, \mathsf{neg}\left(\color{blue}{\frac{154840252661040053153929}{25000000000000000000}}\right)\right)}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\mathsf{fma}\left(\frac{10838835996651139530249}{625000000000000000000}, x \cdot x, \color{blue}{\frac{-154840252661040053153929}{25000000000000000000}}\right)}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      15. sub-neg N/A

          \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\mathsf{fma}\left(\frac{10838835996651139530249}{625000000000000000000}, x \cdot x, \frac{-154840252661040053153929}{25000000000000000000}\right)}{\color{blue}{x \cdot \frac{104109730557}{25000000000} + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000}\right)\right)}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\mathsf{fma}\left(\frac{10838835996651139530249}{625000000000000000000}, x \cdot x, \frac{-154840252661040053153929}{25000000000000000000}\right)}{\color{blue}{x \cdot \frac{104109730557}{25000000000}} + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000}\right)\right)} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      17. *-commutative N/A

          \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\mathsf{fma}\left(\frac{10838835996651139530249}{625000000000000000000}, x \cdot x, \frac{-154840252661040053153929}{25000000000000000000}\right)}{\color{blue}{\frac{104109730557}{25000000000} \cdot x} + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000}\right)\right)} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      18. lower-fma.f64 N/A

          \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\mathsf{fma}\left(\frac{10838835996651139530249}{625000000000000000000}, x \cdot x, \frac{-154840252661040053153929}{25000000000000000000}\right)}{\color{blue}{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \mathsf{neg}\left(\frac{393497462077}{5000000000}\right)\right)}} \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      19. metadata-eval 99.1

          \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\frac{\mathsf{fma}\left(17.342137594641823, x \cdot x, -6193.6101064416025\right)}{\mathsf{fma}\left(4.16438922228, x, \color{blue}{-78.6994924154}\right)} \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\left(\color{blue}{\frac{\mathsf{fma}\left(17.342137594641823, x \cdot x, -6193.6101064416025\right)}{\mathsf{fma}\left(4.16438922228, x, -78.6994924154\right)}} \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   if 1.00000000000000006e290 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.2%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right) \cdot \left(-1 \cdot x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right) \cdot \left(-1 \cdot x\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(-4.16438922228 - \frac{-110.1139242984811 - \frac{\left(-3655.1204654076414 + \frac{130977.50649958357}{x}\right) - \frac{y}{x}}{x}}{x}\right) \cdot \left(-x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z + \left(y + \left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x} \leq 10^{+290}:\\ \;\;\;\;\frac{\left(\left(\left(\frac{\mathsf{fma}\left(17.342137594641823, x \cdot x, -6193.6101064416025\right)}{\mathsf{fma}\left(4.16438922228, x, -78.6994924154\right)} \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x\\ \mathbf{if}\;\frac{\left(z + \left(y + \left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x - 2\right)}{t\_0} \leq 10^{+290}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right) \cdot x, x, 137.519416416 \cdot x\right) + y\right) \cdot x + z\right) \cdot \left(x - 2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (+
          47.066876606
          (*
           (+ 313.399215894 (* (+ 263.505074721 (* (+ 43.3400022514 x) x)) x))
           x))))
   (if (<=
        (/
         (*
          (+
           z
           (*
            (+
             y
             (*
              (+ 137.519416416 (* (+ 78.6994924154 (* 4.16438922228 x)) x))
              x))
            x))
          (- x 2.0))
         t_0)
        1e+290)
     (/
      (*
       (+
        (*
         (+
          (fma (* (fma 4.16438922228 x 78.6994924154) x) x (* 137.519416416 x))
          y)
         x)
        z)
       (- x 2.0))
      t_0)
     (*
      (-
       (/
        (-
         -110.1139242984811
         (/ (- (+ (/ 130977.50649958357 x) -3655.1204654076414) (/ y x)) x))
        x)
       -4.16438922228)
      x))))

C

double code(double x, double y, double z) {
	double t_0 = 47.066876606 + ((313.399215894 + ((263.505074721 + ((43.3400022514 + x) * x)) * x)) * x);
	double tmp;
	if ((((z + ((y + ((137.519416416 + ((78.6994924154 + (4.16438922228 * x)) * x)) * x)) * x)) * (x - 2.0)) / t_0) <= 1e+290) {
		tmp = ((((fma((fma(4.16438922228, x, 78.6994924154) * x), x, (137.519416416 * x)) + y) * x) + z) * (x - 2.0)) / t_0;
	} else {
		tmp = (((-110.1139242984811 - ((((130977.50649958357 / x) + -3655.1204654076414) - (y / x)) / x)) / x) - -4.16438922228) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(47.066876606 + Float64(Float64(313.399215894 + Float64(Float64(263.505074721 + Float64(Float64(43.3400022514 + x) * x)) * x)) * x))
	tmp = 0.0
	if (Float64(Float64(Float64(z + Float64(Float64(y + Float64(Float64(137.519416416 + Float64(Float64(78.6994924154 + Float64(4.16438922228 * x)) * x)) * x)) * x)) * Float64(x - 2.0)) / t_0) <= 1e+290)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(Float64(fma(4.16438922228, x, 78.6994924154) * x), x, Float64(137.519416416 * x)) + y) * x) + z) * Float64(x - 2.0)) / t_0);
	else
		tmp = Float64(Float64(Float64(Float64(-110.1139242984811 - Float64(Float64(Float64(Float64(130977.50649958357 / x) + -3655.1204654076414) - Float64(y / x)) / x)) / x) - -4.16438922228) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(47.066876606 + N[(N[(313.399215894 + N[(N[(263.505074721 + N[(N[(43.3400022514 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(z + N[(N[(y + N[(N[(137.519416416 + N[(N[(78.6994924154 + N[(4.16438922228 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], 1e+290], N[(N[(N[(N[(N[(N[(N[(N[(4.16438922228 * x + 78.6994924154), $MachinePrecision] * x), $MachinePrecision] * x + N[(137.519416416 * x), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision] + z), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[(N[(-110.1139242984811 - N[(N[(N[(N[(130977.50649958357 / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64))) < 1.00000000000000006e290

   1. Initial program 99.0%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\color{blue}{\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x} + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\color{blue}{x \cdot \left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right)} + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(x \cdot \color{blue}{\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right)} + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\color{blue}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x\right) \cdot x + \frac{4297481763}{31250000} \cdot x\right)} + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\color{blue}{\mathsf{fma}\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x, x, \frac{4297481763}{31250000} \cdot x\right)} + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\mathsf{fma}\left(\color{blue}{\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right)} \cdot x, x, \frac{4297481763}{31250000} \cdot x\right) + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\mathsf{fma}\left(\left(\color{blue}{x \cdot \frac{104109730557}{25000000000}} + \frac{393497462077}{5000000000}\right) \cdot x, x, \frac{4297481763}{31250000} \cdot x\right) + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\mathsf{fma}\left(\left(\color{blue}{\frac{104109730557}{25000000000} \cdot x} + \frac{393497462077}{5000000000}\right) \cdot x, x, \frac{4297481763}{31250000} \cdot x\right) + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{393497462077}{5000000000}\right)} \cdot x, x, \frac{4297481763}{31250000} \cdot x\right) + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      10. lower-*.f64 99.1

          \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right) \cdot x, x, \color{blue}{137.519416416 \cdot x}\right) + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   4. Applied rewrites 99.1%

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right) \cdot x, x, 137.519416416 \cdot x\right)} + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   if 1.00000000000000006e290 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.2%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right) \cdot \left(-1 \cdot x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right) \cdot \left(-1 \cdot x\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(-4.16438922228 - \frac{-110.1139242984811 - \frac{\left(-3655.1204654076414 + \frac{130977.50649958357}{x}\right) - \frac{y}{x}}{x}}{x}\right) \cdot \left(-x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z + \left(y + \left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x} \leq 10^{+290}:\\ \;\;\;\;\frac{\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right) \cdot x, x, 137.519416416 \cdot x\right) + y\right) \cdot x + z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(z + \left(y + \left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x} \leq 10^{+290}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right) \cdot \left(x - 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (+
         z
         (*
          (+
           y
           (* (+ 137.519416416 (* (+ 78.6994924154 (* 4.16438922228 x)) x)) x))
          x))
        (- x 2.0))
       (+
        47.066876606
        (*
         (+ 313.399215894 (* (+ 263.505074721 (* (+ 43.3400022514 x) x)) x))
         x)))
      1e+290)
   (/
    (*
     (fma
      (fma (fma (fma 4.16438922228 x 78.6994924154) x 137.519416416) x y)
      x
      z)
     (- x 2.0))
    (fma
     (fma (fma (+ 43.3400022514 x) x 263.505074721) x 313.399215894)
     x
     47.066876606))
   (*
    (-
     (/
      (-
       -110.1139242984811
       (/ (- (+ (/ 130977.50649958357 x) -3655.1204654076414) (/ y x)) x))
      x)
     -4.16438922228)
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((z + ((y + ((137.519416416 + ((78.6994924154 + (4.16438922228 * x)) * x)) * x)) * x)) * (x - 2.0)) / (47.066876606 + ((313.399215894 + ((263.505074721 + ((43.3400022514 + x) * x)) * x)) * x))) <= 1e+290) {
		tmp = (fma(fma(fma(fma(4.16438922228, x, 78.6994924154), x, 137.519416416), x, y), x, z) * (x - 2.0)) / fma(fma(fma((43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606);
	} else {
		tmp = (((-110.1139242984811 - ((((130977.50649958357 / x) + -3655.1204654076414) - (y / x)) / x)) / x) - -4.16438922228) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(z + Float64(Float64(y + Float64(Float64(137.519416416 + Float64(Float64(78.6994924154 + Float64(4.16438922228 * x)) * x)) * x)) * x)) * Float64(x - 2.0)) / Float64(47.066876606 + Float64(Float64(313.399215894 + Float64(Float64(263.505074721 + Float64(Float64(43.3400022514 + x) * x)) * x)) * x))) <= 1e+290)
		tmp = Float64(Float64(fma(fma(fma(fma(4.16438922228, x, 78.6994924154), x, 137.519416416), x, y), x, z) * Float64(x - 2.0)) / fma(fma(fma(Float64(43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606));
	else
		tmp = Float64(Float64(Float64(Float64(-110.1139242984811 - Float64(Float64(Float64(Float64(130977.50649958357 / x) + -3655.1204654076414) - Float64(y / x)) / x)) / x) - -4.16438922228) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(z + N[(N[(y + N[(N[(137.519416416 + N[(N[(78.6994924154 + N[(4.16438922228 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(47.066876606 + N[(N[(313.399215894 + N[(N[(263.505074721 + N[(N[(43.3400022514 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+290], N[(N[(N[(N[(N[(N[(4.16438922228 * x + 78.6994924154), $MachinePrecision] * x + 137.519416416), $MachinePrecision] * x + y), $MachinePrecision] * x + z), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x + 263.505074721), $MachinePrecision] * x + 313.399215894), $MachinePrecision] * x + 47.066876606), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-110.1139242984811 - N[(N[(N[(N[(130977.50649958357 / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64))) < 1.00000000000000006e290

   1. Initial program 99.0%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}} \]

   if 1.00000000000000006e290 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.2%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right) \cdot \left(-1 \cdot x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right) \cdot \left(-1 \cdot x\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(-4.16438922228 - \frac{-110.1139242984811 - \frac{\left(-3655.1204654076414 + \frac{130977.50649958357}{x}\right) - \frac{y}{x}}{x}}{x}\right) \cdot \left(-x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z + \left(y + \left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x} \leq 10^{+290}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right) \cdot \left(x - 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \mathbf{if}\;x \leq -8.8 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          (-
           (/
            (-
             -110.1139242984811
             (/
              (- (+ (/ 130977.50649958357 x) -3655.1204654076414) (/ y x))
              x))
            x)
           -4.16438922228)
          x)))
   (if (<= x -8.8e+14)
     t_0
     (if (<= x 2.8e+36)
       (/
        (* (fma (fma 137.519416416 x y) x z) (- x 2.0))
        (+
         47.066876606
         (*
          (+ 313.399215894 (* (+ 263.505074721 (* (+ 43.3400022514 x) x)) x))
          x)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((-110.1139242984811 - ((((130977.50649958357 / x) + -3655.1204654076414) - (y / x)) / x)) / x) - -4.16438922228) * x;
	double tmp;
	if (x <= -8.8e+14) {
		tmp = t_0;
	} else if (x <= 2.8e+36) {
		tmp = (fma(fma(137.519416416, x, y), x, z) * (x - 2.0)) / (47.066876606 + ((313.399215894 + ((263.505074721 + ((43.3400022514 + x) * x)) * x)) * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(-110.1139242984811 - Float64(Float64(Float64(Float64(130977.50649958357 / x) + -3655.1204654076414) - Float64(y / x)) / x)) / x) - -4.16438922228) * x)
	tmp = 0.0
	if (x <= -8.8e+14)
		tmp = t_0;
	elseif (x <= 2.8e+36)
		tmp = Float64(Float64(fma(fma(137.519416416, x, y), x, z) * Float64(x - 2.0)) / Float64(47.066876606 + Float64(Float64(313.399215894 + Float64(Float64(263.505074721 + Float64(Float64(43.3400022514 + x) * x)) * x)) * x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(-110.1139242984811 - N[(N[(N[(N[(130977.50649958357 / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -8.8e+14], t$95$0, If[LessEqual[x, 2.8e+36], N[(N[(N[(N[(137.519416416 * x + y), $MachinePrecision] * x + z), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(47.066876606 + N[(N[(313.399215894 + N[(N[(263.505074721 + N[(N[(43.3400022514 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.8e14 or 2.8000000000000001e36 < x

   1. Initial program 9.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right) \cdot \left(-1 \cdot x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right) \cdot \left(-1 \cdot x\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\left(-4.16438922228 - \frac{-110.1139242984811 - \frac{\left(-3655.1204654076414 + \frac{130977.50649958357}{x}\right) - \frac{y}{x}}{x}}{x}\right) \cdot \left(-x\right)} \]

   if -8.8e14 < x < 2.8000000000000001e36

   1. Initial program 99.7%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \color{blue}{\left(z + x \cdot \left(y + \frac{4297481763}{31250000} \cdot x\right)\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \color{blue}{\left(x \cdot \left(y + \frac{4297481763}{31250000} \cdot x\right) + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\color{blue}{\left(y + \frac{4297481763}{31250000} \cdot x\right) \cdot x} + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \color{blue}{\mathsf{fma}\left(y + \frac{4297481763}{31250000} \cdot x, x, z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{4297481763}{31250000} \cdot x + y}, x, z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      5. lower-fma.f64 97.9

         \[\leadsto \frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(137.519416416, x, y\right)}, x, z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   5. Applied rewrites 97.9%

      \[\leadsto \frac{\left(x - 2\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right)}}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.8 \cdot 10^{+14}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 2}{0.24013125253755718}\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x 2.0) 0.24013125253755718)))
   (if (<= x -5.2e+23)
     t_0
     (if (<= x 4e+39)
       (/
        (* (fma (fma 137.519416416 x y) x z) (- x 2.0))
        (+
         47.066876606
         (*
          (+ 313.399215894 (* (+ 263.505074721 (* (+ 43.3400022514 x) x)) x))
          x)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - 2.0) / 0.24013125253755718;
	double tmp;
	if (x <= -5.2e+23) {
		tmp = t_0;
	} else if (x <= 4e+39) {
		tmp = (fma(fma(137.519416416, x, y), x, z) * (x - 2.0)) / (47.066876606 + ((313.399215894 + ((263.505074721 + ((43.3400022514 + x) * x)) * x)) * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - 2.0) / 0.24013125253755718)
	tmp = 0.0
	if (x <= -5.2e+23)
		tmp = t_0;
	elseif (x <= 4e+39)
		tmp = Float64(Float64(fma(fma(137.519416416, x, y), x, z) * Float64(x - 2.0)) / Float64(47.066876606 + Float64(Float64(313.399215894 + Float64(Float64(263.505074721 + Float64(Float64(43.3400022514 + x) * x)) * x)) * x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - 2.0), $MachinePrecision] / 0.24013125253755718), $MachinePrecision]}, If[LessEqual[x, -5.2e+23], t$95$0, If[LessEqual[x, 4e+39], N[(N[(N[(N[(137.519416416 * x + y), $MachinePrecision] * x + z), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(47.066876606 + N[(N[(313.399215894 + N[(N[(263.505074721 + N[(N[(43.3400022514 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999983e23 or 3.99999999999999976e39 < x

   1. Initial program 9.0%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 14.1

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 14.1%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.4%

      \[\leadsto \frac{x - 2}{\color{blue}{0.24013125253755718}} \]

   if -5.19999999999999983e23 < x < 3.99999999999999976e39

   1. Initial program 99.1%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \color{blue}{\left(z + x \cdot \left(y + \frac{4297481763}{31250000} \cdot x\right)\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \color{blue}{\left(x \cdot \left(y + \frac{4297481763}{31250000} \cdot x\right) + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\color{blue}{\left(y + \frac{4297481763}{31250000} \cdot x\right) \cdot x} + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \color{blue}{\mathsf{fma}\left(y + \frac{4297481763}{31250000} \cdot x, x, z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{4297481763}{31250000} \cdot x + y}, x, z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      5. lower-fma.f64 97.3

         \[\leadsto \frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(137.519416416, x, y\right)}, x, z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   5. Applied rewrites 97.3%

      \[\leadsto \frac{\left(x - 2\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right)}}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 2}{0.24013125253755718}\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z\right)}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \mathsf{fma}\left(x, x, -4\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x 2.0) 0.24013125253755718)))
   (if (<= x -5.6e+23)
     t_0
     (if (<= x 4e+39)
       (*
        (/
         (fma y x z)
         (*
          (- x -2.0)
          (fma
           (fma (fma (+ 43.3400022514 x) x 263.505074721) x 313.399215894)
           x
           47.066876606)))
        (fma x x -4.0))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - 2.0) / 0.24013125253755718;
	double tmp;
	if (x <= -5.6e+23) {
		tmp = t_0;
	} else if (x <= 4e+39) {
		tmp = (fma(y, x, z) / ((x - -2.0) * fma(fma(fma((43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606))) * fma(x, x, -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - 2.0) / 0.24013125253755718)
	tmp = 0.0
	if (x <= -5.6e+23)
		tmp = t_0;
	elseif (x <= 4e+39)
		tmp = Float64(Float64(fma(y, x, z) / Float64(Float64(x - -2.0) * fma(fma(fma(Float64(43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606))) * fma(x, x, -4.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - 2.0), $MachinePrecision] / 0.24013125253755718), $MachinePrecision]}, If[LessEqual[x, -5.6e+23], t$95$0, If[LessEqual[x, 4e+39], N[(N[(N[(y * x + z), $MachinePrecision] / N[(N[(x - -2.0), $MachinePrecision] * N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x + 263.505074721), $MachinePrecision] * x + 313.399215894), $MachinePrecision] * x + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x * x + -4.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.6e23 or 3.99999999999999976e39 < x

   1. Initial program 9.0%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 14.1

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 14.1%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.4%

      \[\leadsto \frac{x - 2}{\color{blue}{0.24013125253755718}} \]

   if -5.6e23 < x < 3.99999999999999976e39

   1. Initial program 99.1%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 98.7

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{104109730557}{25000000000} \cdot x + \frac{393497462077}{5000000000}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot \frac{104109730557}{25000000000}} + \frac{393497462077}{5000000000}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      3. flip-+ N/A

         \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{\left(x \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) - \frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}}{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{\frac{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}}{\left(x \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) - \frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}}}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{\frac{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}}{\left(x \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) - \frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}}}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\color{blue}{\frac{x \cdot \frac{104109730557}{25000000000} - \frac{393497462077}{5000000000}}{\left(x \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) - \frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}}}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      7. sub-neg N/A

         \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\color{blue}{x \cdot \frac{104109730557}{25000000000} + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000}\right)\right)}}{\left(x \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) - \frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      8. *-commutative N/A

         \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\color{blue}{\frac{104109730557}{25000000000} \cdot x} + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000}\right)\right)}{\left(x \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) - \frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\color{blue}{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \mathsf{neg}\left(\frac{393497462077}{5000000000}\right)\right)}}{\left(x \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) - \frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \color{blue}{\frac{-393497462077}{5000000000}}\right)}{\left(x \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) - \frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      11. sub-neg N/A

          \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{-393497462077}{5000000000}\right)}{\color{blue}{\left(x \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      12. *-commutative N/A

          \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{-393497462077}{5000000000}\right)}{\color{blue}{\left(\frac{104109730557}{25000000000} \cdot x\right)} \cdot \left(x \cdot \frac{104109730557}{25000000000}\right) + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      13. *-commutative N/A

          \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{-393497462077}{5000000000}\right)}{\left(\frac{104109730557}{25000000000} \cdot x\right) \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      14. swap-sqr N/A

          \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{-393497462077}{5000000000}\right)}{\color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{104109730557}{25000000000}\right) \cdot \left(x \cdot x\right)} + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      15. metadata-eval N/A

          \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{-393497462077}{5000000000}\right)}{\color{blue}{\frac{10838835996651139530249}{625000000000000000000}} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      16. metadata-eval N/A

          \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{-393497462077}{5000000000}\right)}{\color{blue}{\left(\frac{-104109730557}{25000000000} \cdot \frac{-104109730557}{25000000000}\right)} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{-393497462077}{5000000000}\right)}{\color{blue}{\mathsf{fma}\left(\frac{-104109730557}{25000000000} \cdot \frac{-104109730557}{25000000000}, x \cdot x, \mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      18. metadata-eval N/A

          \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{-393497462077}{5000000000}\right)}{\mathsf{fma}\left(\color{blue}{\frac{10838835996651139530249}{625000000000000000000}}, x \cdot x, \mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      19. lower-*.f64 N/A

          \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{-393497462077}{5000000000}\right)}{\mathsf{fma}\left(\frac{10838835996651139530249}{625000000000000000000}, \color{blue}{x \cdot x}, \mathsf{neg}\left(\frac{393497462077}{5000000000} \cdot \frac{393497462077}{5000000000}\right)\right)}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      20. metadata-eval N/A

          \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{-393497462077}{5000000000}\right)}{\mathsf{fma}\left(\frac{10838835996651139530249}{625000000000000000000}, x \cdot x, \mathsf{neg}\left(\color{blue}{\frac{154840252661040053153929}{25000000000000000000}}\right)\right)}}, x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)}} \]

      21. metadata-eval 98.7

          \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{\frac{\mathsf{fma}\left(4.16438922228, x, -78.6994924154\right)}{\mathsf{fma}\left(17.342137594641823, x \cdot x, \color{blue}{-6193.6101064416025}\right)}}, x, 137.519416416\right), x, y\right), x, z\right)}} \]

   6. Applied rewrites 98.7%

      \[\leadsto \frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(4.16438922228, x, -78.6994924154\right)}{\mathsf{fma}\left(17.342137594641823, x \cdot x, -6193.6101064416025\right)}}}, x, 137.519416416\right), x, y\right), x, z\right)}} \]

   8. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}} \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, x, -4\right) \cdot \frac{\color{blue}{z + x \cdot y}}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, -4\right) \cdot \frac{\color{blue}{x \cdot y + z}}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)} \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(x, x, -4\right) \cdot \frac{\color{blue}{y \cdot x} + z}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{216700011257}{5000000000} + x, x, \frac{263505074721}{1000000000}\right), x, \frac{156699607947}{500000000}\right), x, \frac{23533438303}{500000000}\right)} \]

       3. lower-fma.f64 95.5

          \[\leadsto \mathsf{fma}\left(x, x, -4\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(y, x, z\right)}}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \]

   11. Applied rewrites 95.5%

       \[\leadsto \mathsf{fma}\left(x, x, -4\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(y, x, z\right)}}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z\right)}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \mathsf{fma}\left(x, x, -4\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+104}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq -36:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 4.16438922228, x, y - 130977.50649958357\right)}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.14:\\ \;\;\;\;\mathsf{fma}\left(-0.0760470460642314, x, 0.0106231820773988\right) \cdot \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{5.86923874282773}{x} + 0.24013125253755718}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e+104)
   (/ (- x 2.0) 0.24013125253755718)
   (if (<= x -36.0)
     (/ (fma (* (* x x) 4.16438922228) x (- y 130977.50649958357)) (* x x))
     (if (<= x 0.14)
       (*
        (fma -0.0760470460642314 x 0.0106231820773988)
        (*
         (fma x x -4.0)
         (fma
          (fma (fma (fma 4.16438922228 x 78.6994924154) x 137.519416416) x y)
          x
          z)))
       (/ (- x 2.0) (+ (/ 5.86923874282773 x) 0.24013125253755718))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e+104) {
		tmp = (x - 2.0) / 0.24013125253755718;
	} else if (x <= -36.0) {
		tmp = fma(((x * x) * 4.16438922228), x, (y - 130977.50649958357)) / (x * x);
	} else if (x <= 0.14) {
		tmp = fma(-0.0760470460642314, x, 0.0106231820773988) * (fma(x, x, -4.0) * fma(fma(fma(fma(4.16438922228, x, 78.6994924154), x, 137.519416416), x, y), x, z));
	} else {
		tmp = (x - 2.0) / ((5.86923874282773 / x) + 0.24013125253755718);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e+104)
		tmp = Float64(Float64(x - 2.0) / 0.24013125253755718);
	elseif (x <= -36.0)
		tmp = Float64(fma(Float64(Float64(x * x) * 4.16438922228), x, Float64(y - 130977.50649958357)) / Float64(x * x));
	elseif (x <= 0.14)
		tmp = Float64(fma(-0.0760470460642314, x, 0.0106231820773988) * Float64(fma(x, x, -4.0) * fma(fma(fma(fma(4.16438922228, x, 78.6994924154), x, 137.519416416), x, y), x, z)));
	else
		tmp = Float64(Float64(x - 2.0) / Float64(Float64(5.86923874282773 / x) + 0.24013125253755718));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2e+104], N[(N[(x - 2.0), $MachinePrecision] / 0.24013125253755718), $MachinePrecision], If[LessEqual[x, -36.0], N[(N[(N[(N[(x * x), $MachinePrecision] * 4.16438922228), $MachinePrecision] * x + N[(y - 130977.50649958357), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.14], N[(N[(-0.0760470460642314 * x + 0.0106231820773988), $MachinePrecision] * N[(N[(x * x + -4.0), $MachinePrecision] * N[(N[(N[(N[(4.16438922228 * x + 78.6994924154), $MachinePrecision] * x + 137.519416416), $MachinePrecision] * x + y), $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 2.0), $MachinePrecision] / N[(N[(5.86923874282773 / x), $MachinePrecision] + 0.24013125253755718), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2e104

   1. Initial program 0.0%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 0.0

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

      \[\leadsto \frac{x - 2}{\color{blue}{0.24013125253755718}} \]

   if -2e104 < x < -36

   1. Initial program 49.9%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right) \cdot \left(-1 \cdot x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right) \cdot \left(-1 \cdot x\right)} \]

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\left(-4.16438922228 - \frac{-110.1139242984811 - \frac{\left(-3655.1204654076414 + \frac{130977.50649958357}{x}\right) - \frac{y}{x}}{x}}{x}\right) \cdot \left(-x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{-1 \cdot \left(\frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} - y\right) + x \cdot \left(\frac{2284450290879775841688574159837293}{625000000000000000000000000000} + x \cdot \left(\frac{104109730557}{25000000000} \cdot x - \frac{13764240537310136880149}{125000000000000000000}\right)\right)}{\color{blue}{{x}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, -110.1139242984811\right), x, 3655.1204654076414\right), x, -\left(130977.50649958357 - y\right)\right)}{\color{blue}{x \cdot x}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{104109730557}{25000000000} \cdot {x}^{2}, x, -\left(\frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} - y\right)\right)}{x \cdot x} \]
   10. Step-by-step derivation

   11. Applied rewrites 93.9%

       \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 4.16438922228, x, -\left(130977.50649958357 - y\right)\right)}{x \cdot x} \]

   if -36 < x < 0.14000000000000001

   1. Initial program 99.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \frac{1}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{393497462077}{5000000000}\right), x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)\right) \cdot \color{blue}{\left(\frac{250000000}{23533438303} + \frac{-42116581774625000000}{553822718361107519809} \cdot x\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{393497462077}{5000000000}\right), x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)\right) \cdot \color{blue}{\left(\frac{-42116581774625000000}{553822718361107519809} \cdot x + \frac{250000000}{23533438303}\right)} \]

      2. lower-fma.f64 98.8

         \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0760470460642314, x, 0.0106231820773988\right)} \]

   7. Applied rewrites 98.8%

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-0.0760470460642314, x, 0.0106231820773988\right)} \]

   if 0.14000000000000001 < x

   1. Initial program 16.9%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 24.3

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 24.3%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557} + \frac{63615716158700684400745}{10838835996651139530249} \cdot \frac{1}{x}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{63615716158700684400745}{10838835996651139530249} \cdot \frac{1}{x} + \frac{25000000000}{104109730557}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{63615716158700684400745}{10838835996651139530249} \cdot \frac{1}{x} + \frac{25000000000}{104109730557}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\frac{63615716158700684400745}{10838835996651139530249} \cdot 1}{x}} + \frac{25000000000}{104109730557}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x - 2}{\frac{\color{blue}{\frac{63615716158700684400745}{10838835996651139530249}}}{x} + \frac{25000000000}{104109730557}} \]

      5. lower-/.f64 87.9

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{5.86923874282773}{x}} + 0.24013125253755718} \]

   7. Applied rewrites 87.9%

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{5.86923874282773}{x} + 0.24013125253755718}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+104}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq -36:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 4.16438922228, x, y - 130977.50649958357\right)}{x \cdot x}\\ \mathbf{elif}\;x \leq 0.14:\\ \;\;\;\;\mathsf{fma}\left(-0.0760470460642314, x, 0.0106231820773988\right) \cdot \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{5.86923874282773}{x} + 0.24013125253755718}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 2}{0.24013125253755718}\\ \mathbf{if}\;x \leq -5.2 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x 2.0) 0.24013125253755718)))
   (if (<= x -5.2e+23)
     t_0
     (if (<= x 4e+39)
       (/
        (* (fma y x z) (- x 2.0))
        (+
         47.066876606
         (*
          (+ 313.399215894 (* (+ 263.505074721 (* (+ 43.3400022514 x) x)) x))
          x)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - 2.0) / 0.24013125253755718;
	double tmp;
	if (x <= -5.2e+23) {
		tmp = t_0;
	} else if (x <= 4e+39) {
		tmp = (fma(y, x, z) * (x - 2.0)) / (47.066876606 + ((313.399215894 + ((263.505074721 + ((43.3400022514 + x) * x)) * x)) * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - 2.0) / 0.24013125253755718)
	tmp = 0.0
	if (x <= -5.2e+23)
		tmp = t_0;
	elseif (x <= 4e+39)
		tmp = Float64(Float64(fma(y, x, z) * Float64(x - 2.0)) / Float64(47.066876606 + Float64(Float64(313.399215894 + Float64(Float64(263.505074721 + Float64(Float64(43.3400022514 + x) * x)) * x)) * x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - 2.0), $MachinePrecision] / 0.24013125253755718), $MachinePrecision]}, If[LessEqual[x, -5.2e+23], t$95$0, If[LessEqual[x, 4e+39], N[(N[(N[(y * x + z), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(47.066876606 + N[(N[(313.399215894 + N[(N[(263.505074721 + N[(N[(43.3400022514 + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999983e23 or 3.99999999999999976e39 < x

   1. Initial program 9.0%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 14.1

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 14.1%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 95.4%

      \[\leadsto \frac{x - 2}{\color{blue}{0.24013125253755718}} \]

   if -5.19999999999999983e23 < x < 3.99999999999999976e39

   1. Initial program 99.1%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \color{blue}{\left(z + x \cdot y\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \color{blue}{\left(x \cdot y + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(x - 2\right) \cdot \left(\color{blue}{y \cdot x} + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. lower-fma.f64 95.5

         \[\leadsto \frac{\left(x - 2\right) \cdot \color{blue}{\mathsf{fma}\left(y, x, z\right)}}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]

   5. Applied rewrites 95.5%

      \[\leadsto \frac{\left(x - 2\right) \cdot \color{blue}{\mathsf{fma}\left(y, x, z\right)}}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(263.505074721 + \left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+104}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq -36:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 4.16438922228, x, y - 130977.50649958357\right)}{x \cdot x}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+14}:\\ \;\;\;\;0.0106231820773988 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \mathsf{fma}\left(x, x, -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{5.86923874282773}{x} + 0.24013125253755718}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e+104)
   (/ (- x 2.0) 0.24013125253755718)
   (if (<= x -36.0)
     (/ (fma (* (* x x) 4.16438922228) x (- y 130977.50649958357)) (* x x))
     (if (<= x 3e+14)
       (*
        0.0106231820773988
        (* (fma (fma 137.519416416 x y) x z) (fma x x -4.0)))
       (/ (- x 2.0) (+ (/ 5.86923874282773 x) 0.24013125253755718))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e+104) {
		tmp = (x - 2.0) / 0.24013125253755718;
	} else if (x <= -36.0) {
		tmp = fma(((x * x) * 4.16438922228), x, (y - 130977.50649958357)) / (x * x);
	} else if (x <= 3e+14) {
		tmp = 0.0106231820773988 * (fma(fma(137.519416416, x, y), x, z) * fma(x, x, -4.0));
	} else {
		tmp = (x - 2.0) / ((5.86923874282773 / x) + 0.24013125253755718);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e+104)
		tmp = Float64(Float64(x - 2.0) / 0.24013125253755718);
	elseif (x <= -36.0)
		tmp = Float64(fma(Float64(Float64(x * x) * 4.16438922228), x, Float64(y - 130977.50649958357)) / Float64(x * x));
	elseif (x <= 3e+14)
		tmp = Float64(0.0106231820773988 * Float64(fma(fma(137.519416416, x, y), x, z) * fma(x, x, -4.0)));
	else
		tmp = Float64(Float64(x - 2.0) / Float64(Float64(5.86923874282773 / x) + 0.24013125253755718));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2e+104], N[(N[(x - 2.0), $MachinePrecision] / 0.24013125253755718), $MachinePrecision], If[LessEqual[x, -36.0], N[(N[(N[(N[(x * x), $MachinePrecision] * 4.16438922228), $MachinePrecision] * x + N[(y - 130977.50649958357), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e+14], N[(0.0106231820773988 * N[(N[(N[(137.519416416 * x + y), $MachinePrecision] * x + z), $MachinePrecision] * N[(x * x + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 2.0), $MachinePrecision] / N[(N[(5.86923874282773 / x), $MachinePrecision] + 0.24013125253755718), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2e104

   1. Initial program 0.0%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 0.0

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.6%

      \[\leadsto \frac{x - 2}{\color{blue}{0.24013125253755718}} \]

   if -2e104 < x < -36

   1. Initial program 49.9%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right) \cdot \left(-1 \cdot x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right) \cdot \left(-1 \cdot x\right)} \]

   5. Applied rewrites 93.7%

      \[\leadsto \color{blue}{\left(-4.16438922228 - \frac{-110.1139242984811 - \frac{\left(-3655.1204654076414 + \frac{130977.50649958357}{x}\right) - \frac{y}{x}}{x}}{x}\right) \cdot \left(-x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{-1 \cdot \left(\frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} - y\right) + x \cdot \left(\frac{2284450290879775841688574159837293}{625000000000000000000000000000} + x \cdot \left(\frac{104109730557}{25000000000} \cdot x - \frac{13764240537310136880149}{125000000000000000000}\right)\right)}{\color{blue}{{x}^{2}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, -110.1139242984811\right), x, 3655.1204654076414\right), x, -\left(130977.50649958357 - y\right)\right)}{\color{blue}{x \cdot x}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\frac{104109730557}{25000000000} \cdot {x}^{2}, x, -\left(\frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} - y\right)\right)}{x \cdot x} \]
   10. Step-by-step derivation

   11. Applied rewrites 93.9%

       \[\leadsto \frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 4.16438922228, x, -\left(130977.50649958357 - y\right)\right)}{x \cdot x} \]

   if -36 < x < 3e14

   1. Initial program 99.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \frac{1}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{393497462077}{5000000000}\right), x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)\right) \cdot \color{blue}{\frac{250000000}{23533438303}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.9%

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \color{blue}{0.0106231820773988} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{4297481763}{31250000}}, x, y\right), x, z\right)\right) \cdot \frac{250000000}{23533438303} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.9%

       \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{137.519416416}, x, y\right), x, z\right)\right) \cdot 0.0106231820773988 \]

   if 3e14 < x

   1. Initial program 14.4%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 21.9

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 21.9%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557} + \frac{63615716158700684400745}{10838835996651139530249} \cdot \frac{1}{x}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{63615716158700684400745}{10838835996651139530249} \cdot \frac{1}{x} + \frac{25000000000}{104109730557}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{63615716158700684400745}{10838835996651139530249} \cdot \frac{1}{x} + \frac{25000000000}{104109730557}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\frac{63615716158700684400745}{10838835996651139530249} \cdot 1}{x}} + \frac{25000000000}{104109730557}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x - 2}{\frac{\color{blue}{\frac{63615716158700684400745}{10838835996651139530249}}}{x} + \frac{25000000000}{104109730557}} \]

      5. lower-/.f64 90.6

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{5.86923874282773}{x}} + 0.24013125253755718} \]

   7. Applied rewrites 90.6%

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{5.86923874282773}{x} + 0.24013125253755718}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+104}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq -36:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 4.16438922228, x, y - 130977.50649958357\right)}{x \cdot x}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+14}:\\ \;\;\;\;0.0106231820773988 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \mathsf{fma}\left(x, x, -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{5.86923874282773}{x} + 0.24013125253755718}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -36:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+14}:\\ \;\;\;\;0.0106231820773988 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \mathsf{fma}\left(x, x, -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{5.86923874282773}{x} + 0.24013125253755718}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -36.0)
   (/ (- x 2.0) 0.24013125253755718)
   (if (<= x 3e+14)
     (*
      0.0106231820773988
      (* (fma (fma 137.519416416 x y) x z) (fma x x -4.0)))
     (/ (- x 2.0) (+ (/ 5.86923874282773 x) 0.24013125253755718)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -36.0) {
		tmp = (x - 2.0) / 0.24013125253755718;
	} else if (x <= 3e+14) {
		tmp = 0.0106231820773988 * (fma(fma(137.519416416, x, y), x, z) * fma(x, x, -4.0));
	} else {
		tmp = (x - 2.0) / ((5.86923874282773 / x) + 0.24013125253755718);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -36.0)
		tmp = Float64(Float64(x - 2.0) / 0.24013125253755718);
	elseif (x <= 3e+14)
		tmp = Float64(0.0106231820773988 * Float64(fma(fma(137.519416416, x, y), x, z) * fma(x, x, -4.0)));
	else
		tmp = Float64(Float64(x - 2.0) / Float64(Float64(5.86923874282773 / x) + 0.24013125253755718));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -36.0], N[(N[(x - 2.0), $MachinePrecision] / 0.24013125253755718), $MachinePrecision], If[LessEqual[x, 3e+14], N[(0.0106231820773988 * N[(N[(N[(137.519416416 * x + y), $MachinePrecision] * x + z), $MachinePrecision] * N[(x * x + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - 2.0), $MachinePrecision] / N[(N[(5.86923874282773 / x), $MachinePrecision] + 0.24013125253755718), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -36

   1. Initial program 15.5%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 17.1

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 89.5%

      \[\leadsto \frac{x - 2}{\color{blue}{0.24013125253755718}} \]

   if -36 < x < 3e14

   1. Initial program 99.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \frac{1}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{393497462077}{5000000000}\right), x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)\right) \cdot \color{blue}{\frac{250000000}{23533438303}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.9%

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \color{blue}{0.0106231820773988} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{4297481763}{31250000}}, x, y\right), x, z\right)\right) \cdot \frac{250000000}{23533438303} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.9%

       \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{137.519416416}, x, y\right), x, z\right)\right) \cdot 0.0106231820773988 \]

   if 3e14 < x

   1. Initial program 14.4%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 21.9

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 21.9%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557} + \frac{63615716158700684400745}{10838835996651139530249} \cdot \frac{1}{x}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{63615716158700684400745}{10838835996651139530249} \cdot \frac{1}{x} + \frac{25000000000}{104109730557}}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{63615716158700684400745}{10838835996651139530249} \cdot \frac{1}{x} + \frac{25000000000}{104109730557}}} \]

      3. associate-*r/ N/A

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\frac{63615716158700684400745}{10838835996651139530249} \cdot 1}{x}} + \frac{25000000000}{104109730557}} \]

      4. metadata-eval N/A

         \[\leadsto \frac{x - 2}{\frac{\color{blue}{\frac{63615716158700684400745}{10838835996651139530249}}}{x} + \frac{25000000000}{104109730557}} \]

      5. lower-/.f64 90.6

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{5.86923874282773}{x}} + 0.24013125253755718} \]

   7. Applied rewrites 90.6%

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{5.86923874282773}{x} + 0.24013125253755718}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -36:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+14}:\\ \;\;\;\;0.0106231820773988 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \mathsf{fma}\left(x, x, -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{\frac{5.86923874282773}{x} + 0.24013125253755718}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 2}{0.24013125253755718}\\ \mathbf{if}\;x \leq -36:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+14}:\\ \;\;\;\;0.0106231820773988 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \mathsf{fma}\left(x, x, -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x 2.0) 0.24013125253755718)))
   (if (<= x -36.0)
     t_0
     (if (<= x 3e+14)
       (*
        0.0106231820773988
        (* (fma (fma 137.519416416 x y) x z) (fma x x -4.0)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - 2.0) / 0.24013125253755718;
	double tmp;
	if (x <= -36.0) {
		tmp = t_0;
	} else if (x <= 3e+14) {
		tmp = 0.0106231820773988 * (fma(fma(137.519416416, x, y), x, z) * fma(x, x, -4.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - 2.0) / 0.24013125253755718)
	tmp = 0.0
	if (x <= -36.0)
		tmp = t_0;
	elseif (x <= 3e+14)
		tmp = Float64(0.0106231820773988 * Float64(fma(fma(137.519416416, x, y), x, z) * fma(x, x, -4.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - 2.0), $MachinePrecision] / 0.24013125253755718), $MachinePrecision]}, If[LessEqual[x, -36.0], t$95$0, If[LessEqual[x, 3e+14], N[(0.0106231820773988 * N[(N[(N[(137.519416416 * x + y), $MachinePrecision] * x + z), $MachinePrecision] * N[(x * x + -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -36 or 3e14 < x

   1. Initial program 14.9%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 19.6

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.0%

      \[\leadsto \frac{x - 2}{\color{blue}{0.24013125253755718}} \]

   if -36 < x < 3e14

   1. Initial program 99.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \frac{1}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{393497462077}{5000000000}\right), x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)\right) \cdot \color{blue}{\frac{250000000}{23533438303}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.9%

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \color{blue}{0.0106231820773988} \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{4297481763}{31250000}}, x, y\right), x, z\right)\right) \cdot \frac{250000000}{23533438303} \]
   9. Step-by-step derivation

   10. Applied rewrites 96.9%

       \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{137.519416416}, x, y\right), x, z\right)\right) \cdot 0.0106231820773988 \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -36:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+14}:\\ \;\;\;\;0.0106231820773988 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \mathsf{fma}\left(x, x, -4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 2}{0.24013125253755718}\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, 0.3041881842569256 \cdot z\right), x, -0.0424927283095952 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x 2.0) 0.24013125253755718)))
   (if (<= x -2.65e+22)
     t_0
     (if (<= x 3e+14)
       (fma
        (fma -0.0424927283095952 y (* 0.3041881842569256 z))
        x
        (* -0.0424927283095952 z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - 2.0) / 0.24013125253755718;
	double tmp;
	if (x <= -2.65e+22) {
		tmp = t_0;
	} else if (x <= 3e+14) {
		tmp = fma(fma(-0.0424927283095952, y, (0.3041881842569256 * z)), x, (-0.0424927283095952 * z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - 2.0) / 0.24013125253755718)
	tmp = 0.0
	if (x <= -2.65e+22)
		tmp = t_0;
	elseif (x <= 3e+14)
		tmp = fma(fma(-0.0424927283095952, y, Float64(0.3041881842569256 * z)), x, Float64(-0.0424927283095952 * z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - 2.0), $MachinePrecision] / 0.24013125253755718), $MachinePrecision]}, If[LessEqual[x, -2.65e+22], t$95$0, If[LessEqual[x, 3e+14], N[(N[(-0.0424927283095952 * y + N[(0.3041881842569256 * z), $MachinePrecision]), $MachinePrecision] * x + N[(-0.0424927283095952 * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6499999999999999e22 or 3e14 < x

   1. Initial program 13.6%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 18.4

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 18.4%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 92.3%

      \[\leadsto \frac{x - 2}{\color{blue}{0.24013125253755718}} \]

   if -2.6499999999999999e22 < x < 3e14

   1. Initial program 99.1%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   4. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \frac{1}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right) + \frac{-1000000000}{23533438303} \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right) \cdot x} + \frac{-1000000000}{23533438303} \cdot z \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z, x, \frac{-1000000000}{23533438303} \cdot z\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1000000000}{23533438303} \cdot y + \left(\mathsf{neg}\left(\frac{-168466327098500000000}{553822718361107519809} \cdot z\right)\right)}, x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \mathsf{neg}\left(\frac{-168466327098500000000}{553822718361107519809} \cdot z\right)\right)}, x, \frac{-1000000000}{23533438303} \cdot z\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{\left(\mathsf{neg}\left(\frac{-168466327098500000000}{553822718361107519809}\right)\right) \cdot z}\right), x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{\frac{168466327098500000000}{553822718361107519809}} \cdot z\right), x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{\frac{168466327098500000000}{553822718361107519809} \cdot z}\right), x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      9. lower-*.f64 93.4

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, 0.3041881842569256 \cdot z\right), x, \color{blue}{-0.0424927283095952 \cdot z}\right) \]

   7. Applied rewrites 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, 0.3041881842569256 \cdot z\right), x, -0.0424927283095952 \cdot z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 88.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 2}{0.24013125253755718}\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.95:\\ \;\;\;\;\left(\mathsf{fma}\left(y, x, z\right) \cdot -4\right) \cdot 0.0106231820773988\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x 2.0) 0.24013125253755718)))
   (if (<= x -2.65e+22)
     t_0
     (if (<= x 1.95) (* (* (fma y x z) -4.0) 0.0106231820773988) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - 2.0) / 0.24013125253755718;
	double tmp;
	if (x <= -2.65e+22) {
		tmp = t_0;
	} else if (x <= 1.95) {
		tmp = (fma(y, x, z) * -4.0) * 0.0106231820773988;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - 2.0) / 0.24013125253755718)
	tmp = 0.0
	if (x <= -2.65e+22)
		tmp = t_0;
	elseif (x <= 1.95)
		tmp = Float64(Float64(fma(y, x, z) * -4.0) * 0.0106231820773988);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - 2.0), $MachinePrecision] / 0.24013125253755718), $MachinePrecision]}, If[LessEqual[x, -2.65e+22], t$95$0, If[LessEqual[x, 1.95], N[(N[(N[(y * x + z), $MachinePrecision] * -4.0), $MachinePrecision] * 0.0106231820773988), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6499999999999999e22 or 1.94999999999999996 < x

   1. Initial program 15.0%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 19.7

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 19.7%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.8%

      \[\leadsto \frac{x - 2}{\color{blue}{0.24013125253755718}} \]

   if -2.6499999999999999e22 < x < 1.94999999999999996

   1. Initial program 99.1%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   4. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \frac{1}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{104109730557}{25000000000}, x, \frac{393497462077}{5000000000}\right), x, \frac{4297481763}{31250000}\right), x, y\right), x, z\right)\right) \cdot \color{blue}{\frac{250000000}{23533438303}} \]
   6. Step-by-step derivation

   7. Applied rewrites 96.0%

      \[\leadsto \left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \color{blue}{0.0106231820773988} \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(-4 \cdot z + -4 \cdot \left(x \cdot y\right)\right)} \cdot \frac{250000000}{23533438303} \]
   9. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(z + x \cdot y\right)\right)} \cdot \frac{250000000}{23533438303} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-4 \cdot \left(z + x \cdot y\right)\right)} \cdot \frac{250000000}{23533438303} \]

      3. +-commutative N/A

         \[\leadsto \left(-4 \cdot \color{blue}{\left(x \cdot y + z\right)}\right) \cdot \frac{250000000}{23533438303} \]

      4. *-commutative N/A

         \[\leadsto \left(-4 \cdot \left(\color{blue}{y \cdot x} + z\right)\right) \cdot \frac{250000000}{23533438303} \]

      5. lower-fma.f64 94.3

         \[\leadsto \left(-4 \cdot \color{blue}{\mathsf{fma}\left(y, x, z\right)}\right) \cdot 0.0106231820773988 \]

   10. Applied rewrites 94.3%

       \[\leadsto \color{blue}{\left(-4 \cdot \mathsf{fma}\left(y, x, z\right)\right)} \cdot 0.0106231820773988 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+22}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \mathbf{elif}\;x \leq 1.95:\\ \;\;\;\;\left(\mathsf{fma}\left(y, x, z\right) \cdot -4\right) \cdot 0.0106231820773988\\ \mathbf{else}:\\ \;\;\;\;\frac{x - 2}{0.24013125253755718}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 2}{0.24013125253755718}\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.95:\\ \;\;\;\;\mathsf{fma}\left(-0.0424927283095952 \cdot y, x, -0.0424927283095952 \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x 2.0) 0.24013125253755718)))
   (if (<= x -2.65e+22)
     t_0
     (if (<= x 1.95)
       (fma (* -0.0424927283095952 y) x (* -0.0424927283095952 z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - 2.0) / 0.24013125253755718;
	double tmp;
	if (x <= -2.65e+22) {
		tmp = t_0;
	} else if (x <= 1.95) {
		tmp = fma((-0.0424927283095952 * y), x, (-0.0424927283095952 * z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - 2.0) / 0.24013125253755718)
	tmp = 0.0
	if (x <= -2.65e+22)
		tmp = t_0;
	elseif (x <= 1.95)
		tmp = fma(Float64(-0.0424927283095952 * y), x, Float64(-0.0424927283095952 * z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - 2.0), $MachinePrecision] / 0.24013125253755718), $MachinePrecision]}, If[LessEqual[x, -2.65e+22], t$95$0, If[LessEqual[x, 1.95], N[(N[(-0.0424927283095952 * y), $MachinePrecision] * x + N[(-0.0424927283095952 * z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6499999999999999e22 or 1.94999999999999996 < x

   1. Initial program 15.0%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 19.7

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 19.7%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.8%

      \[\leadsto \frac{x - 2}{\color{blue}{0.24013125253755718}} \]

   if -2.6499999999999999e22 < x < 1.94999999999999996

   1. Initial program 99.1%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   4. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \frac{1}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right) + \frac{-1000000000}{23533438303} \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right) \cdot x} + \frac{-1000000000}{23533438303} \cdot z \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z, x, \frac{-1000000000}{23533438303} \cdot z\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1000000000}{23533438303} \cdot y + \left(\mathsf{neg}\left(\frac{-168466327098500000000}{553822718361107519809} \cdot z\right)\right)}, x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \mathsf{neg}\left(\frac{-168466327098500000000}{553822718361107519809} \cdot z\right)\right)}, x, \frac{-1000000000}{23533438303} \cdot z\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{\left(\mathsf{neg}\left(\frac{-168466327098500000000}{553822718361107519809}\right)\right) \cdot z}\right), x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{\frac{168466327098500000000}{553822718361107519809}} \cdot z\right), x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{\frac{168466327098500000000}{553822718361107519809} \cdot z}\right), x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      9. lower-*.f64 94.7

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, 0.3041881842569256 \cdot z\right), x, \color{blue}{-0.0424927283095952 \cdot z}\right) \]

   7. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, 0.3041881842569256 \cdot z\right), x, -0.0424927283095952 \cdot z\right)} \]

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303} \cdot y, x, \frac{-1000000000}{23533438303} \cdot z\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 94.3%

       \[\leadsto \mathsf{fma}\left(-0.0424927283095952 \cdot y, x, -0.0424927283095952 \cdot z\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 76.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 2}{0.24013125253755718}\\ \mathbf{if}\;x \leq -62000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(0.3041881842569256, x, -0.0424927283095952\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- x 2.0) 0.24013125253755718)))
   (if (<= x -62000.0)
     t_0
     (if (<= x 3e+14)
       (* (fma 0.3041881842569256 x -0.0424927283095952) z)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x - 2.0) / 0.24013125253755718;
	double tmp;
	if (x <= -62000.0) {
		tmp = t_0;
	} else if (x <= 3e+14) {
		tmp = fma(0.3041881842569256, x, -0.0424927283095952) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x - 2.0) / 0.24013125253755718)
	tmp = 0.0
	if (x <= -62000.0)
		tmp = t_0;
	elseif (x <= 3e+14)
		tmp = Float64(fma(0.3041881842569256, x, -0.0424927283095952) * z);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x - 2.0), $MachinePrecision] / 0.24013125253755718), $MachinePrecision]}, If[LessEqual[x, -62000.0], t$95$0, If[LessEqual[x, 3e+14], N[(N[(0.3041881842569256 * x + -0.0424927283095952), $MachinePrecision] * z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -62000 or 3e14 < x

   1. Initial program 14.2%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z\right)}}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(x - 2\right) \cdot \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}} \]

      4. clear-num N/A

         \[\leadsto \left(x - 2\right) \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x - 2}{\frac{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}}} \]

      7. lower-/.f64 19.0

         \[\leadsto \frac{x - 2}{\color{blue}{\frac{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}}} \]

   4. Applied rewrites 19.0%

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{x - 2}{\color{blue}{\frac{25000000000}{104109730557}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.8%

      \[\leadsto \frac{x - 2}{\color{blue}{0.24013125253755718}} \]

   if -62000 < x < 3e14

   1. Initial program 99.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \frac{1}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right) + \frac{-1000000000}{23533438303} \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right) \cdot x} + \frac{-1000000000}{23533438303} \cdot z \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z, x, \frac{-1000000000}{23533438303} \cdot z\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1000000000}{23533438303} \cdot y + \left(\mathsf{neg}\left(\frac{-168466327098500000000}{553822718361107519809} \cdot z\right)\right)}, x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \mathsf{neg}\left(\frac{-168466327098500000000}{553822718361107519809} \cdot z\right)\right)}, x, \frac{-1000000000}{23533438303} \cdot z\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{\left(\mathsf{neg}\left(\frac{-168466327098500000000}{553822718361107519809}\right)\right) \cdot z}\right), x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{\frac{168466327098500000000}{553822718361107519809}} \cdot z\right), x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{\frac{168466327098500000000}{553822718361107519809} \cdot z}\right), x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      9. lower-*.f64 94.7

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, 0.3041881842569256 \cdot z\right), x, \color{blue}{-0.0424927283095952 \cdot z}\right) \]

   7. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, 0.3041881842569256 \cdot z\right), x, -0.0424927283095952 \cdot z\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto z \cdot \color{blue}{\left(\frac{168466327098500000000}{553822718361107519809} \cdot x - \frac{1000000000}{23533438303}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 70.5%

       \[\leadsto \mathsf{fma}\left(0.3041881842569256, x, -0.0424927283095952\right) \cdot \color{blue}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 76.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-11}:\\ \;\;\;\;4.16438922228 \cdot x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(0.3041881842569256, x, -0.0424927283095952\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.6e-11)
   (* 4.16438922228 x)
   (if (<= x 3e+14)
     (* (fma 0.3041881842569256 x -0.0424927283095952) z)
     (* 4.16438922228 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6e-11) {
		tmp = 4.16438922228 * x;
	} else if (x <= 3e+14) {
		tmp = fma(0.3041881842569256, x, -0.0424927283095952) * z;
	} else {
		tmp = 4.16438922228 * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.6e-11)
		tmp = Float64(4.16438922228 * x);
	elseif (x <= 3e+14)
		tmp = Float64(fma(0.3041881842569256, x, -0.0424927283095952) * z);
	else
		tmp = Float64(4.16438922228 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.6e-11], N[(4.16438922228 * x), $MachinePrecision], If[LessEqual[x, 3e+14], N[(N[(0.3041881842569256 * x + -0.0424927283095952), $MachinePrecision] * z), $MachinePrecision], N[(4.16438922228 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.59999999999999997e-11 or 3e14 < x

   1. Initial program 15.6%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 89.0

         \[\leadsto \color{blue}{4.16438922228 \cdot x} \]

   5. Applied rewrites 89.0%

      \[\leadsto \color{blue}{4.16438922228 \cdot x} \]

   if -1.59999999999999997e-11 < x < 3e14

   1. Initial program 99.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x, -4\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)\right) \cdot \frac{1}{\left(x - -2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right) + \frac{-1000000000}{23533438303} \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right) \cdot x} + \frac{-1000000000}{23533438303} \cdot z \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z, x, \frac{-1000000000}{23533438303} \cdot z\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1000000000}{23533438303} \cdot y + \left(\mathsf{neg}\left(\frac{-168466327098500000000}{553822718361107519809} \cdot z\right)\right)}, x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \mathsf{neg}\left(\frac{-168466327098500000000}{553822718361107519809} \cdot z\right)\right)}, x, \frac{-1000000000}{23533438303} \cdot z\right) \]

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

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{\left(\mathsf{neg}\left(\frac{-168466327098500000000}{553822718361107519809}\right)\right) \cdot z}\right), x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{\frac{168466327098500000000}{553822718361107519809}} \cdot z\right), x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{\frac{168466327098500000000}{553822718361107519809} \cdot z}\right), x, \frac{-1000000000}{23533438303} \cdot z\right) \]

      9. lower-*.f64 96.0

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, 0.3041881842569256 \cdot z\right), x, \color{blue}{-0.0424927283095952 \cdot z}\right) \]

   7. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, 0.3041881842569256 \cdot z\right), x, -0.0424927283095952 \cdot z\right)} \]

   8. Taylor expanded in z around inf

      \[\leadsto z \cdot \color{blue}{\left(\frac{168466327098500000000}{553822718361107519809} \cdot x - \frac{1000000000}{23533438303}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 71.4%

       \[\leadsto \mathsf{fma}\left(0.3041881842569256, x, -0.0424927283095952\right) \cdot \color{blue}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 76.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -62000:\\ \;\;\;\;4.16438922228 \cdot x\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;-0.0424927283095952 \cdot z\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -62000.0)
   (* 4.16438922228 x)
   (if (<= x 2.0) (* -0.0424927283095952 z) (* 4.16438922228 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -62000.0) {
		tmp = 4.16438922228 * x;
	} else if (x <= 2.0) {
		tmp = -0.0424927283095952 * z;
	} else {
		tmp = 4.16438922228 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-62000.0d0)) then
        tmp = 4.16438922228d0 * x
    else if (x <= 2.0d0) then
        tmp = (-0.0424927283095952d0) * z
    else
        tmp = 4.16438922228d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -62000.0) {
		tmp = 4.16438922228 * x;
	} else if (x <= 2.0) {
		tmp = -0.0424927283095952 * z;
	} else {
		tmp = 4.16438922228 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -62000.0:
		tmp = 4.16438922228 * x
	elif x <= 2.0:
		tmp = -0.0424927283095952 * z
	else:
		tmp = 4.16438922228 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -62000.0)
		tmp = Float64(4.16438922228 * x);
	elseif (x <= 2.0)
		tmp = Float64(-0.0424927283095952 * z);
	else
		tmp = Float64(4.16438922228 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -62000.0)
		tmp = 4.16438922228 * x;
	elseif (x <= 2.0)
		tmp = -0.0424927283095952 * z;
	else
		tmp = 4.16438922228 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -62000.0], N[(4.16438922228 * x), $MachinePrecision], If[LessEqual[x, 2.0], N[(-0.0424927283095952 * z), $MachinePrecision], N[(4.16438922228 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -62000 or 2 < x

   1. Initial program 15.6%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x} \]
   4. Step-by-step derivation

      1. lower-*.f64 88.9

         \[\leadsto \color{blue}{4.16438922228 \cdot x} \]

   5. Applied rewrites 88.9%

      \[\leadsto \color{blue}{4.16438922228 \cdot x} \]

   if -62000 < x < 2

   1. Initial program 99.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z} \]
   4. Step-by-step derivation

      1. lower-*.f64 71.0

         \[\leadsto \color{blue}{-0.0424927283095952 \cdot z} \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{-0.0424927283095952 \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 34.3% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ -0.0424927283095952 \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* -0.0424927283095952 z))

C

double code(double x, double y, double z) {
	return -0.0424927283095952 * z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (-0.0424927283095952d0) * z
end function

Java

public static double code(double x, double y, double z) {
	return -0.0424927283095952 * z;
}

Python

def code(x, y, z):
	return -0.0424927283095952 * z

Julia

function code(x, y, z)
	return Float64(-0.0424927283095952 * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -0.0424927283095952 * z;
end

Wolfram

code[x_, y_, z_] := N[(-0.0424927283095952 * z), $MachinePrecision]

Derivation

1. Initial program 59.7%

   \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z} \]
4. Step-by-step derivation

   1. lower-*.f64 38.5

      \[\leadsto \color{blue}{-0.0424927283095952 \cdot z} \]

5. Applied rewrites 38.5%

   \[\leadsto \color{blue}{-0.0424927283095952 \cdot z} \]
6. Add Preprocessing

Developer Target 1: 98.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{if}\;x < -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))
   (if (< x -3.326128725870005e+62)
     t_0
     (if (< x 9.429991714554673e+55)
       (*
        (/ (- x 2.0) 1.0)
        (/
         (+
          (*
           (+
            (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x)
            y)
           x)
          z)
         (+
          (*
           (+
            (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x))))
            313.399215894)
           x)
          47.066876606)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / (x * x)) + (4.16438922228 * x)) - 110.1139242984811;
	double tmp;
	if (x < -3.326128725870005e+62) {
		tmp = t_0;
	} else if (x < 9.429991714554673e+55) {
		tmp = ((x - 2.0) / 1.0) * (((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z) / (((((263.505074721 * x) + ((43.3400022514 * (x * x)) + (x * (x * x)))) + 313.399215894) * x) + 47.066876606));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / (x * x)) + (4.16438922228d0 * x)) - 110.1139242984811d0
    if (x < (-3.326128725870005d+62)) then
        tmp = t_0
    else if (x < 9.429991714554673d+55) then
        tmp = ((x - 2.0d0) / 1.0d0) * (((((((((x * 4.16438922228d0) + 78.6994924154d0) * x) + 137.519416416d0) * x) + y) * x) + z) / (((((263.505074721d0 * x) + ((43.3400022514d0 * (x * x)) + (x * (x * x)))) + 313.399215894d0) * x) + 47.066876606d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / (x * x)) + (4.16438922228 * x)) - 110.1139242984811;
	double tmp;
	if (x < -3.326128725870005e+62) {
		tmp = t_0;
	} else if (x < 9.429991714554673e+55) {
		tmp = ((x - 2.0) / 1.0) * (((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z) / (((((263.505074721 * x) + ((43.3400022514 * (x * x)) + (x * (x * x)))) + 313.399215894) * x) + 47.066876606));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / (x * x)) + (4.16438922228 * x)) - 110.1139242984811
	tmp = 0
	if x < -3.326128725870005e+62:
		tmp = t_0
	elif x < 9.429991714554673e+55:
		tmp = ((x - 2.0) / 1.0) * (((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z) / (((((263.505074721 * x) + ((43.3400022514 * (x * x)) + (x * (x * x)))) + 313.399215894) * x) + 47.066876606))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / Float64(x * x)) + Float64(4.16438922228 * x)) - 110.1139242984811)
	tmp = 0.0
	if (x < -3.326128725870005e+62)
		tmp = t_0;
	elseif (x < 9.429991714554673e+55)
		tmp = Float64(Float64(Float64(x - 2.0) / 1.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z) / Float64(Float64(Float64(Float64(Float64(263.505074721 * x) + Float64(Float64(43.3400022514 * Float64(x * x)) + Float64(x * Float64(x * x)))) + 313.399215894) * x) + 47.066876606)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / (x * x)) + (4.16438922228 * x)) - 110.1139242984811;
	tmp = 0.0;
	if (x < -3.326128725870005e+62)
		tmp = t_0;
	elseif (x < 9.429991714554673e+55)
		tmp = ((x - 2.0) / 1.0) * (((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z) / (((((263.505074721 * x) + ((43.3400022514 * (x * x)) + (x * (x * x)))) + 313.399215894) * x) + 47.066876606));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(4.16438922228 * x), $MachinePrecision]), $MachinePrecision] - 110.1139242984811), $MachinePrecision]}, If[Less[x, -3.326128725870005e+62], t$95$0, If[Less[x, 9.429991714554673e+55], N[(N[(N[(x - 2.0), $MachinePrecision] / 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision] * x), $MachinePrecision] + 137.519416416), $MachinePrecision] * x), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision] + z), $MachinePrecision] / N[(N[(N[(N[(N[(263.505074721 * x), $MachinePrecision] + N[(N[(43.3400022514 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision] * x), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024249 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -332612872587000500000000000000000000000000000000000000000000000) (- (+ (/ y (* x x)) (* 104109730557/25000000000 x)) 1101139242984811/10000000000000) (if (< x 94299917145546730000000000000000000000000000000000000000) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (+ (* (+ (+ (* 263505074721/1000000000 x) (+ (* 216700011257/5000000000 (* x x)) (* x (* x x)))) 156699607947/500000000) x) 23533438303/500000000))) (- (+ (/ y (* x x)) (* 104109730557/25000000000 x)) 1101139242984811/10000000000000))))

  (/ (* (- x 2.0) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))