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

Percentage Accurate: 58.0% → 98.3%

Time: 18.2s

Alternatives: 20

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: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot x, 17.342137594641823, -6193.6101064416025\right)}{\mathsf{fma}\left(x, 4.16438922228, -78.6994924154\right)}, 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          x
          (+
           (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
           y))
         z))
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606))
      INFINITY)
   (/
    (/
     (fma
      x
      (fma
       x
       (fma
        x
        (/
         (fma (* x x) 17.342137594641823 -6193.6101064416025)
         (fma x 4.16438922228 -78.6994924154))
        137.519416416)
       y)
      z)
     (fma
      x
      (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
      47.066876606))
    (/ 1.0 (+ x -2.0)))
   (/ 1.0 (/ 0.24013125253755718 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= ((double) INFINITY)) {
		tmp = (fma(x, fma(x, fma(x, (fma((x * x), 17.342137594641823, -6193.6101064416025) / fma(x, 4.16438922228, -78.6994924154)), 137.519416416), y), z) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)) / (1.0 / (x + -2.0));
	} else {
		tmp = 1.0 / (0.24013125253755718 / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= Inf)
		tmp = Float64(Float64(fma(x, fma(x, fma(x, Float64(fma(Float64(x * x), 17.342137594641823, -6193.6101064416025) / fma(x, 4.16438922228, -78.6994924154)), 137.519416416), y), z) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)) / Float64(1.0 / Float64(x + -2.0)));
	else
		tmp = Float64(1.0 / Float64(0.24013125253755718 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x * N[(x * N[(x * N[(N[(N[(x * x), $MachinePrecision] * 17.342137594641823 + -6193.6101064416025), $MachinePrecision] / N[(x * 4.16438922228 + -78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.24013125253755718 / x), $MachinePrecision]), $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))) < +inf.0

   1. Initial program 92.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. swap-sqr N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. metadata-eval 98.9

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

   6. Applied egg-rr 98.9%

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

   if +inf.0 < (/.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 0.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

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

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{25000000000}{104109730557}}{x}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 99.4

         \[\leadsto \frac{1}{\color{blue}{\frac{0.24013125253755718}{x}}} \]

   9. Simplified 99.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{0.24013125253755718}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x \cdot x, 17.342137594641823, -6193.6101064416025\right)}{\mathsf{fma}\left(x, 4.16438922228, -78.6994924154\right)}, 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          x
          (+
           (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
           y))
         z))
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606))
      INFINITY)
   (/
    (fma
     x
     (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
     z)
    (/
     (fma
      x
      (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
      47.066876606)
     (+ x -2.0)))
   (/ 1.0 (/ 0.24013125253755718 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= ((double) INFINITY)) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / (fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606) / (x + -2.0));
	} else {
		tmp = 1.0 / (0.24013125253755718 / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= Inf)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / Float64(fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606) / Float64(x + -2.0)));
	else
		tmp = Float64(1.0 / Float64(0.24013125253755718 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] / N[(N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision] / N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.24013125253755718 / x), $MachinePrecision]), $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))) < +inf.0

   1. Initial program 92.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. *-commutative N/A

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

      3. div-inv N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 98.9%

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

   if +inf.0 < (/.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 0.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

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

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{25000000000}{104109730557}}{x}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 99.4

         \[\leadsto \frac{1}{\color{blue}{\frac{0.24013125253755718}{x}}} \]

   9. Simplified 99.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{0.24013125253755718}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq \infty:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq \infty:\\ \;\;\;\;\frac{x + -2}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          x
          (+
           (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
           y))
         z))
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606))
      INFINITY)
   (/
    (+ x -2.0)
    (/
     (fma
      x
      (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
      47.066876606)
     (fma
      x
      (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
      z)))
   (/ 1.0 (/ 0.24013125253755718 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= ((double) INFINITY)) {
		tmp = (x + -2.0) / (fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606) / fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z));
	} else {
		tmp = 1.0 / (0.24013125253755718 / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= Inf)
		tmp = Float64(Float64(x + -2.0) / Float64(fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606) / fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z)));
	else
		tmp = Float64(1.0 / Float64(0.24013125253755718 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x + -2.0), $MachinePrecision] / N[(N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.24013125253755718 / x), $MachinePrecision]), $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))) < +inf.0

   1. Initial program 92.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. sub-neg N/A

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

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

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

      7. metadata-eval N/A

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

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

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

   4. Applied egg-rr 98.7%

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

   if +inf.0 < (/.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 0.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

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

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{25000000000}{104109730557}}{x}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 99.4

         \[\leadsto \frac{1}{\color{blue}{\frac{0.24013125253755718}{x}}} \]

   9. Simplified 99.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{0.24013125253755718}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq \infty:\\ \;\;\;\;\frac{x + -2}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          x
          (+
           (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
           y))
         z))
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606))
      INFINITY)
   (*
    (fma
     x
     (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
     z)
    (/
     (+ x -2.0)
     (fma
      x
      (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
      47.066876606)))
   (/ 1.0 (/ 0.24013125253755718 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= ((double) INFINITY)) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * ((x + -2.0) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606));
	} else {
		tmp = 1.0 / (0.24013125253755718 / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= Inf)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * Float64(Float64(x + -2.0) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)));
	else
		tmp = Float64(1.0 / Float64(0.24013125253755718 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * N[(N[(x + -2.0), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.24013125253755718 / x), $MachinePrecision]), $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))) < +inf.0

   1. Initial program 92.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 98.6%

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

   if +inf.0 < (/.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 0.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

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

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

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

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{25000000000}{104109730557}}{x}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 99.4

         \[\leadsto \frac{1}{\color{blue}{\frac{0.24013125253755718}{x}}} \]

   9. Simplified 99.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{0.24013125253755718}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{1}{\frac{\frac{6.349501247902845 + \frac{-42.87307123793778}{x}}{x} - -0.24013125253755718}{x}}\\ \mathbf{elif}\;x \leq 9.5:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + x \cdot \mathsf{fma}\left(x, 137.519416416, y\right)\right)}{\mathsf{fma}\left(x, 313.399215894, 47.066876606\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.35)
   (/
    1.0
    (/
     (-
      (/ (+ 6.349501247902845 (/ -42.87307123793778 x)) x)
      -0.24013125253755718)
     x))
   (if (<= x 9.5)
     (/
      (* (- x 2.0) (+ z (* x (fma x 137.519416416 y))))
      (fma x 313.399215894 47.066876606))
     (if (<= x 8.5e+73)
       (/
        1.0
        (/
         (* x (* x x))
         (fma
          x
          (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
          z)))
       (/ 1.0 (/ 0.24013125253755718 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.35) {
		tmp = 1.0 / ((((6.349501247902845 + (-42.87307123793778 / x)) / x) - -0.24013125253755718) / x);
	} else if (x <= 9.5) {
		tmp = ((x - 2.0) * (z + (x * fma(x, 137.519416416, y)))) / fma(x, 313.399215894, 47.066876606);
	} else if (x <= 8.5e+73) {
		tmp = 1.0 / ((x * (x * x)) / fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z));
	} else {
		tmp = 1.0 / (0.24013125253755718 / x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.35)
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(6.349501247902845 + Float64(-42.87307123793778 / x)) / x) - -0.24013125253755718) / x));
	elseif (x <= 9.5)
		tmp = Float64(Float64(Float64(x - 2.0) * Float64(z + Float64(x * fma(x, 137.519416416, y)))) / fma(x, 313.399215894, 47.066876606));
	elseif (x <= 8.5e+73)
		tmp = Float64(1.0 / Float64(Float64(x * Float64(x * x)) / fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z)));
	else
		tmp = Float64(1.0 / Float64(0.24013125253755718 / x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.35], N[(1.0 / N[(N[(N[(N[(6.349501247902845 + N[(-42.87307123793778 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -0.24013125253755718), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(z + N[(x * N[(x * 137.519416416 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 313.399215894 + 47.066876606), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e+73], N[(1.0 / N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(0.24013125253755718 / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.3500000000000001

   1. Initial program 17.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 25.3%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

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

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

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

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

   6. Applied egg-rr 25.3%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{68821202686550684400745}{10838835996651139530249} - \frac{48379186685422091754128006001718670}{1128428295162862690821234941118693} \cdot \frac{1}{x}}{x} - \frac{25000000000}{104109730557}}{x}}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{68821202686550684400745}{10838835996651139530249} - \frac{48379186685422091754128006001718670}{1128428295162862690821234941118693} \cdot \frac{1}{x}}{x} - \frac{25000000000}{104109730557}}{x}\right)}} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot \frac{\frac{68821202686550684400745}{10838835996651139530249} - \frac{48379186685422091754128006001718670}{1128428295162862690821234941118693} \cdot \frac{1}{x}}{x} - \frac{25000000000}{104109730557}}{\mathsf{neg}\left(x\right)}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{\frac{-1 \cdot \frac{\frac{68821202686550684400745}{10838835996651139530249} - \frac{48379186685422091754128006001718670}{1128428295162862690821234941118693} \cdot \frac{1}{x}}{x} - \frac{25000000000}{104109730557}}{\color{blue}{-1 \cdot x}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot \frac{\frac{68821202686550684400745}{10838835996651139530249} - \frac{48379186685422091754128006001718670}{1128428295162862690821234941118693} \cdot \frac{1}{x}}{x} - \frac{25000000000}{104109730557}}{-1 \cdot x}}} \]

   9. Simplified 90.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{6.349501247902845 + \frac{-42.87307123793778}{x}}{-x} + -0.24013125253755718}{-x}}} \]

   if -1.3500000000000001 < x < 9.5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\mathsf{fma}\left(x, \frac{4297481763}{31250000}, y\right) \cdot x + z\right)}{\color{blue}{\frac{23533438303}{500000000} + \frac{156699607947}{500000000} \cdot x}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 98.2

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

   8. Simplified 98.2%

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

   if 9.5 < x < 8.4999999999999998e73

   1. Initial program 65.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 93.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

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

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

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

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

   6. Applied egg-rr 93.9%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{3}}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}} \]
   8. Step-by-step derivation

      1. cube-mult N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot \left(x \cdot x\right)}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{1}{\frac{x \cdot \color{blue}{{x}^{2}}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot {x}^{2}}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}} \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{\frac{x \cdot \color{blue}{\left(x \cdot x\right)}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}} \]

      5. *-lowering-*.f64 86.0

         \[\leadsto \frac{1}{\frac{x \cdot \color{blue}{\left(x \cdot x\right)}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}} \]

   9. Simplified 86.0%

      \[\leadsto \frac{1}{\frac{\color{blue}{x \cdot \left(x \cdot x\right)}}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}} \]

   if 8.4999999999999998e73 < x

   1. Initial program 0.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 1.7%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

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

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

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

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

   6. Applied egg-rr 1.8%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{25000000000}{104109730557}}{x}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 99.4

         \[\leadsto \frac{1}{\color{blue}{\frac{0.24013125253755718}{x}}} \]

   9. Simplified 99.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{0.24013125253755718}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{1}{\frac{\frac{6.349501247902845 + \frac{-42.87307123793778}{x}}{x} - -0.24013125253755718}{x}}\\ \mathbf{elif}\;x \leq 9.5:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + x \cdot \mathsf{fma}\left(x, 137.519416416, y\right)\right)}{\mathsf{fma}\left(x, 313.399215894, 47.066876606\right)}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+73}:\\ \;\;\;\;\frac{1}{\frac{x \cdot \left(x \cdot x\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\frac{\frac{\frac{y + -130977.50649958357}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \mathbf{if}\;x \leq -105000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 140000:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + x \cdot \mathsf{fma}\left(x, 137.519416416, y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), x \cdot x, \mathsf{fma}\left(x, 313.399215894, 47.066876606\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          x
          (-
           (/
            (+
             (/ (- (/ (+ y -130977.50649958357) x) -3655.1204654076414) x)
             -110.1139242984811)
            x)
           -4.16438922228))))
   (if (<= x -105000000000.0)
     t_0
     (if (<= x 140000.0)
       (/
        (* (- x 2.0) (+ z (* x (fma x 137.519416416 y))))
        (fma
         (fma x (+ x 43.3400022514) 263.505074721)
         (* x x)
         (fma x 313.399215894 47.066876606)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (((((((y + -130977.50649958357) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228);
	double tmp;
	if (x <= -105000000000.0) {
		tmp = t_0;
	} else if (x <= 140000.0) {
		tmp = ((x - 2.0) * (z + (x * fma(x, 137.519416416, y)))) / fma(fma(x, (x + 43.3400022514), 263.505074721), (x * x), fma(x, 313.399215894, 47.066876606));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + -130977.50649958357) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228))
	tmp = 0.0
	if (x <= -105000000000.0)
		tmp = t_0;
	elseif (x <= 140000.0)
		tmp = Float64(Float64(Float64(x - 2.0) * Float64(z + Float64(x * fma(x, 137.519416416, y)))) / fma(fma(x, Float64(x + 43.3400022514), 263.505074721), Float64(x * x), fma(x, 313.399215894, 47.066876606)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[(N[(N[(N[(N[(N[(y + -130977.50649958357), $MachinePrecision] / x), $MachinePrecision] - -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + -110.1139242984811), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -105000000000.0], t$95$0, If[LessEqual[x, 140000.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(z + N[(x * N[(x * 137.519416416 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x * 313.399215894 + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e11 or 1.4e5 < x

   1. Initial program 16.7%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 16.0%

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

   6. Taylor expanded in x around -inf

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

   8. Simplified 95.3%

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

   if -1.05e11 < x < 1.4e5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.2

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

   5. Simplified 99.2%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

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

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

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

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

      13. accelerator-lowering-fma.f64 99.2

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

   7. Applied egg-rr 99.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -105000000000:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y + -130977.50649958357}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \mathbf{elif}\;x \leq 140000:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + x \cdot \mathsf{fma}\left(x, 137.519416416, y\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), x \cdot x, \mathsf{fma}\left(x, 313.399215894, 47.066876606\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y + -130977.50649958357}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\frac{\frac{\frac{y + -130977.50649958357}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \mathbf{if}\;x \leq -40000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 140000:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          x
          (-
           (/
            (+
             (/ (- (/ (+ y -130977.50649958357) x) -3655.1204654076414) x)
             -110.1139242984811)
            x)
           -4.16438922228))))
   (if (<= x -40000000000.0)
     t_0
     (if (<= x 140000.0)
       (/
        (* (- x 2.0) (fma x (fma x 137.519416416 y) z))
        (+
         (*
          x
          (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
         47.066876606))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (((((((y + -130977.50649958357) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228);
	double tmp;
	if (x <= -40000000000.0) {
		tmp = t_0;
	} else if (x <= 140000.0) {
		tmp = ((x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + -130977.50649958357) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228))
	tmp = 0.0
	if (x <= -40000000000.0)
		tmp = t_0;
	elseif (x <= 140000.0)
		tmp = Float64(Float64(Float64(x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[(N[(N[(N[(N[(N[(y + -130977.50649958357), $MachinePrecision] / x), $MachinePrecision] - -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + -110.1139242984811), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -40000000000.0], t$95$0, If[LessEqual[x, 140000.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(x * N[(x * 137.519416416 + y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e10 or 1.4e5 < x

   1. Initial program 16.7%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 16.0%

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

   6. Taylor expanded in x around -inf

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

   8. Simplified 95.3%

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

   if -4e10 < x < 1.4e5

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 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. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 99.2

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

   5. Simplified 99.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40000000000:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y + -130977.50649958357}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \mathbf{elif}\;x \leq 140000:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y + -130977.50649958357}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\frac{\frac{\frac{y + -130977.50649958357}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 38:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + x \cdot \mathsf{fma}\left(x, 137.519416416, y\right)\right)}{\mathsf{fma}\left(x, 313.399215894, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          x
          (-
           (/
            (+
             (/ (- (/ (+ y -130977.50649958357) x) -3655.1204654076414) x)
             -110.1139242984811)
            x)
           -4.16438922228))))
   (if (<= x -1.35)
     t_0
     (if (<= x 38.0)
       (/
        (* (- x 2.0) (+ z (* x (fma x 137.519416416 y))))
        (fma x 313.399215894 47.066876606))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (((((((y + -130977.50649958357) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228);
	double tmp;
	if (x <= -1.35) {
		tmp = t_0;
	} else if (x <= 38.0) {
		tmp = ((x - 2.0) * (z + (x * fma(x, 137.519416416, y)))) / fma(x, 313.399215894, 47.066876606);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + -130977.50649958357) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228))
	tmp = 0.0
	if (x <= -1.35)
		tmp = t_0;
	elseif (x <= 38.0)
		tmp = Float64(Float64(Float64(x - 2.0) * Float64(z + Float64(x * fma(x, 137.519416416, y)))) / fma(x, 313.399215894, 47.066876606));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[(N[(N[(N[(N[(N[(y + -130977.50649958357), $MachinePrecision] / x), $MachinePrecision] - -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + -110.1139242984811), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35], t$95$0, If[LessEqual[x, 38.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(z + N[(x * N[(x * 137.519416416 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 313.399215894 + 47.066876606), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3500000000000001 or 38 < x

   1. Initial program 17.4%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 16.7%

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

   6. Taylor expanded in x around -inf

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

   8. Simplified 94.9%

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

   if -1.3500000000000001 < x < 38

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\mathsf{fma}\left(x, \frac{4297481763}{31250000}, y\right) \cdot x + z\right)}{\color{blue}{\frac{23533438303}{500000000} + \frac{156699607947}{500000000} \cdot x}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 98.2

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

   8. Simplified 98.2%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y + -130977.50649958357}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \mathbf{elif}\;x \leq 38:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + x \cdot \mathsf{fma}\left(x, 137.519416416, y\right)\right)}{\mathsf{fma}\left(x, 313.399215894, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y + -130977.50649958357}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{\frac{6.349501247902845 + \frac{-42.87307123793778}{x}}{x} - -0.24013125253755718}{x}}\\ \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 420:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + x \cdot \mathsf{fma}\left(x, 137.519416416, y\right)\right)}{\mathsf{fma}\left(x, 313.399215894, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          1.0
          (/
           (-
            (/ (+ 6.349501247902845 (/ -42.87307123793778 x)) x)
            -0.24013125253755718)
           x))))
   (if (<= x -1.35)
     t_0
     (if (<= x 420.0)
       (/
        (* (- x 2.0) (+ z (* x (fma x 137.519416416 y))))
        (fma x 313.399215894 47.066876606))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 / ((((6.349501247902845 + (-42.87307123793778 / x)) / x) - -0.24013125253755718) / x);
	double tmp;
	if (x <= -1.35) {
		tmp = t_0;
	} else if (x <= 420.0) {
		tmp = ((x - 2.0) * (z + (x * fma(x, 137.519416416, y)))) / fma(x, 313.399215894, 47.066876606);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 / Float64(Float64(Float64(Float64(6.349501247902845 + Float64(-42.87307123793778 / x)) / x) - -0.24013125253755718) / x))
	tmp = 0.0
	if (x <= -1.35)
		tmp = t_0;
	elseif (x <= 420.0)
		tmp = Float64(Float64(Float64(x - 2.0) * Float64(z + Float64(x * fma(x, 137.519416416, y)))) / fma(x, 313.399215894, 47.066876606));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 / N[(N[(N[(N[(6.349501247902845 + N[(-42.87307123793778 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -0.24013125253755718), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35], t$95$0, If[LessEqual[x, 420.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(z + N[(x * N[(x * 137.519416416 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 313.399215894 + 47.066876606), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3500000000000001 or 420 < x

   1. Initial program 17.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 25.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

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

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

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

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

   6. Applied egg-rr 25.5%

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

   7. Taylor expanded in x around -inf

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{68821202686550684400745}{10838835996651139530249} - \frac{48379186685422091754128006001718670}{1128428295162862690821234941118693} \cdot \frac{1}{x}}{x} - \frac{25000000000}{104109730557}}{x}}} \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{1}{\color{blue}{\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{68821202686550684400745}{10838835996651139530249} - \frac{48379186685422091754128006001718670}{1128428295162862690821234941118693} \cdot \frac{1}{x}}{x} - \frac{25000000000}{104109730557}}{x}\right)}} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot \frac{\frac{68821202686550684400745}{10838835996651139530249} - \frac{48379186685422091754128006001718670}{1128428295162862690821234941118693} \cdot \frac{1}{x}}{x} - \frac{25000000000}{104109730557}}{\mathsf{neg}\left(x\right)}}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{1}{\frac{-1 \cdot \frac{\frac{68821202686550684400745}{10838835996651139530249} - \frac{48379186685422091754128006001718670}{1128428295162862690821234941118693} \cdot \frac{1}{x}}{x} - \frac{25000000000}{104109730557}}{\color{blue}{-1 \cdot x}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{-1 \cdot \frac{\frac{68821202686550684400745}{10838835996651139530249} - \frac{48379186685422091754128006001718670}{1128428295162862690821234941118693} \cdot \frac{1}{x}}{x} - \frac{25000000000}{104109730557}}{-1 \cdot x}}} \]

   9. Simplified 89.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{6.349501247902845 + \frac{-42.87307123793778}{x}}{-x} + -0.24013125253755718}{-x}}} \]

   if -1.3500000000000001 < x < 420

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\mathsf{fma}\left(x, \frac{4297481763}{31250000}, y\right) \cdot x + z\right)}{\color{blue}{\frac{23533438303}{500000000} + \frac{156699607947}{500000000} \cdot x}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 98.2

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

   8. Simplified 98.2%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;\frac{1}{\frac{\frac{6.349501247902845 + \frac{-42.87307123793778}{x}}{x} - -0.24013125253755718}{x}}\\ \mathbf{elif}\;x \leq 420:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + x \cdot \mathsf{fma}\left(x, 137.519416416, y\right)\right)}{\mathsf{fma}\left(x, 313.399215894, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{6.349501247902845 + \frac{-42.87307123793778}{x}}{x} - -0.24013125253755718}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \mathbf{if}\;x \leq -37:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5800:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + x \cdot \mathsf{fma}\left(x, 137.519416416, y\right)\right)}{\mathsf{fma}\left(x, 313.399215894, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          x
          (+
           4.16438922228
           (/ (+ -110.1139242984811 (/ 3655.1204654076414 x)) x)))))
   (if (<= x -37.0)
     t_0
     (if (<= x 5800.0)
       (/
        (* (- x 2.0) (+ z (* x (fma x 137.519416416 y))))
        (fma x 313.399215894 47.066876606))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.16438922228 + ((-110.1139242984811 + (3655.1204654076414 / x)) / x));
	double tmp;
	if (x <= -37.0) {
		tmp = t_0;
	} else if (x <= 5800.0) {
		tmp = ((x - 2.0) * (z + (x * fma(x, 137.519416416, y)))) / fma(x, 313.399215894, 47.066876606);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.16438922228 + Float64(Float64(-110.1139242984811 + Float64(3655.1204654076414 / x)) / x)))
	tmp = 0.0
	if (x <= -37.0)
		tmp = t_0;
	elseif (x <= 5800.0)
		tmp = Float64(Float64(Float64(x - 2.0) * Float64(z + Float64(x * fma(x, 137.519416416, y)))) / fma(x, 313.399215894, 47.066876606));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.16438922228 + N[(N[(-110.1139242984811 + N[(3655.1204654076414 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -37.0], t$95$0, If[LessEqual[x, 5800.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(z + N[(x * N[(x * 137.519416416 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 313.399215894 + 47.066876606), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -37 or 5800 < x

   1. Initial program 17.4%

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

   3. Taylor expanded in x around inf

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

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

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}}\right) - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} \]

      2. associate--l+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)\right)} \]

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. /-lowering-/.f64 N/A

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

      12. sub-neg N/A

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\color{blue}{\frac{2284450290879775841688574159837293}{625000000000000000000000000000} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}}{x}\right) \]

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

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\color{blue}{\frac{2284450290879775841688574159837293}{625000000000000000000000000000} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}}{x}\right) \]

      14. associate-*r/ N/A

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\color{blue}{\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000} \cdot 1}{x}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}{x}\right) \]

      15. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\frac{\color{blue}{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}}{x} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}{x}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\color{blue}{\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}{x}\right) \]

      17. metadata-eval 88.8

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

   5. Simplified 88.8%

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

   if -37 < x < 5800

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\mathsf{fma}\left(x, \frac{4297481763}{31250000}, y\right) \cdot x + z\right)}{\color{blue}{\frac{23533438303}{500000000} + \frac{156699607947}{500000000} \cdot x}} \]
   7. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 98.2

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

   8. Simplified 98.2%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -37:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 5800:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + x \cdot \mathsf{fma}\left(x, 137.519416416, y\right)\right)}{\mathsf{fma}\left(x, 313.399215894, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7500000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 2400:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + x \cdot \mathsf{fma}\left(x, 137.519416416, y\right)\right)}{47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7500000.0)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 2400.0)
     (/ (* (- x 2.0) (+ z (* x (fma x 137.519416416 y)))) 47.066876606)
     (*
      x
      (+
       4.16438922228
       (/ (+ -110.1139242984811 (/ 3655.1204654076414 x)) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7500000.0) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 2400.0) {
		tmp = ((x - 2.0) * (z + (x * fma(x, 137.519416416, y)))) / 47.066876606;
	} else {
		tmp = x * (4.16438922228 + ((-110.1139242984811 + (3655.1204654076414 / x)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7500000.0)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 2400.0)
		tmp = Float64(Float64(Float64(x - 2.0) * Float64(z + Float64(x * fma(x, 137.519416416, y)))) / 47.066876606);
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(Float64(-110.1139242984811 + Float64(3655.1204654076414 / x)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7500000.0], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 2400.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(z + N[(x * N[(x * 137.519416416 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 47.066876606), $MachinePrecision], N[(x * N[(4.16438922228 + N[(N[(-110.1139242984811 + N[(3655.1204654076414 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e6

   1. Initial program 16.2%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 12.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000} \cdot 1}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\color{blue}{\frac{104109730557}{25000000000}}}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000}}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot 1}{x \cdot z}}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

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

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\color{blue}{\frac{-13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\frac{-13764240537310136880149}{125000000000000000000}}{\color{blue}{z \cdot x}}\right)\right) \]

      13. *-lowering-*.f64 64.2

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

   8. Simplified 64.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x - \frac{13764240537310136880149}{125000000000000000000}} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{-13764240537310136880149}{125000000000000000000}} \]

       4. accelerator-lowering-fma.f64 91.8

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   11. Simplified 91.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   if -7.5e6 < x < 2400

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.2

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(\mathsf{fma}\left(x, \frac{4297481763}{31250000}, y\right) \cdot x + z\right)}{\color{blue}{\frac{23533438303}{500000000}}} \]
   7. Step-by-step derivation

   8. Simplified 96.3%

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

   if 2400 < x

   1. Initial program 17.1%

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

   3. Taylor expanded in x around inf

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

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

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}}\right) - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} \]

      2. associate--l+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)\right)} \]

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. /-lowering-/.f64 N/A

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

      12. sub-neg N/A

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\color{blue}{\frac{2284450290879775841688574159837293}{625000000000000000000000000000} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}}{x}\right) \]

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

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\color{blue}{\frac{2284450290879775841688574159837293}{625000000000000000000000000000} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}}{x}\right) \]

      14. associate-*r/ N/A

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\color{blue}{\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000} \cdot 1}{x}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}{x}\right) \]

      15. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\frac{\color{blue}{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}}{x} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}{x}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\color{blue}{\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}{x}\right) \]

      17. metadata-eval 87.9

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

   5. Simplified 87.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7500000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 2400:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(z + x \cdot \mathsf{fma}\left(x, 137.519416416, y\right)\right)}{47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7500000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 7:\\ \;\;\;\;\mathsf{fma}\left(-0.0424927283095952, z, x \cdot \mathsf{fma}\left(z, 0.28294182010212804, \mathsf{fma}\left(z, 0.0212463641547976, y \cdot -0.0424927283095952\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7500000.0)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 7.0)
     (fma
      -0.0424927283095952
      z
      (*
       x
       (fma
        z
        0.28294182010212804
        (fma z 0.0212463641547976 (* y -0.0424927283095952)))))
     (*
      x
      (+
       4.16438922228
       (/ (+ -110.1139242984811 (/ 3655.1204654076414 x)) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7500000.0) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 7.0) {
		tmp = fma(-0.0424927283095952, z, (x * fma(z, 0.28294182010212804, fma(z, 0.0212463641547976, (y * -0.0424927283095952)))));
	} else {
		tmp = x * (4.16438922228 + ((-110.1139242984811 + (3655.1204654076414 / x)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7500000.0)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 7.0)
		tmp = fma(-0.0424927283095952, z, Float64(x * fma(z, 0.28294182010212804, fma(z, 0.0212463641547976, Float64(y * -0.0424927283095952)))));
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(Float64(-110.1139242984811 + Float64(3655.1204654076414 / x)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7500000.0], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 7.0], N[(-0.0424927283095952 * z + N[(x * N[(z * 0.28294182010212804 + N[(z * 0.0212463641547976 + N[(y * -0.0424927283095952), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(4.16438922228 + N[(N[(-110.1139242984811 + N[(3655.1204654076414 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e6

   1. Initial program 16.2%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 12.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000} \cdot 1}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\color{blue}{\frac{104109730557}{25000000000}}}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000}}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot 1}{x \cdot z}}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

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

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\color{blue}{\frac{-13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\frac{-13764240537310136880149}{125000000000000000000}}{\color{blue}{z \cdot x}}\right)\right) \]

      13. *-lowering-*.f64 64.2

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

   8. Simplified 64.2%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x - \frac{13764240537310136880149}{125000000000000000000}} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{-13764240537310136880149}{125000000000000000000}} \]

       4. accelerator-lowering-fma.f64 91.8

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   11. Simplified 91.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   if -7.5e6 < x < 7

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. swap-sqr N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. metadata-eval 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)} \]
   8. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right)} \]

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

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, \color{blue}{x \cdot \left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \color{blue}{\left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) + \left(\mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right) + \frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot \frac{-156699607947000000000}{553822718361107519809}}\right)\right) + \frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809}\right)\right)} + \frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \color{blue}{\mathsf{fma}\left(z, \mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809}\right), \frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right)\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \mathsf{fma}\left(z, \color{blue}{\frac{156699607947000000000}{553822718361107519809}}, \frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right)\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \mathsf{fma}\left(z, \frac{156699607947000000000}{553822718361107519809}, \color{blue}{\frac{500000000}{23533438303} \cdot z + \frac{500000000}{23533438303} \cdot \left(-2 \cdot y\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \mathsf{fma}\left(z, \frac{156699607947000000000}{553822718361107519809}, \color{blue}{z \cdot \frac{500000000}{23533438303}} + \frac{500000000}{23533438303} \cdot \left(-2 \cdot y\right)\right)\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \mathsf{fma}\left(z, \frac{156699607947000000000}{553822718361107519809}, \color{blue}{\mathsf{fma}\left(z, \frac{500000000}{23533438303}, \frac{500000000}{23533438303} \cdot \left(-2 \cdot y\right)\right)}\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \mathsf{fma}\left(z, \frac{156699607947000000000}{553822718361107519809}, \mathsf{fma}\left(z, \frac{500000000}{23533438303}, \color{blue}{\left(\frac{500000000}{23533438303} \cdot -2\right) \cdot y}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \mathsf{fma}\left(z, \frac{156699607947000000000}{553822718361107519809}, \mathsf{fma}\left(z, \frac{500000000}{23533438303}, \color{blue}{\frac{-1000000000}{23533438303}} \cdot y\right)\right)\right) \]

      14. *-lowering-*.f64 92.7

          \[\leadsto \mathsf{fma}\left(-0.0424927283095952, z, x \cdot \mathsf{fma}\left(z, 0.28294182010212804, \mathsf{fma}\left(z, 0.0212463641547976, \color{blue}{-0.0424927283095952 \cdot y}\right)\right)\right) \]

   9. Simplified 92.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.0424927283095952, z, x \cdot \mathsf{fma}\left(z, 0.28294182010212804, \mathsf{fma}\left(z, 0.0212463641547976, -0.0424927283095952 \cdot y\right)\right)\right)} \]

   if 7 < x

   1. Initial program 17.1%

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

   3. Taylor expanded in x around inf

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

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

         \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}}\right) - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} \]

      2. associate--l+ N/A

         \[\leadsto x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)\right)} \]

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. /-lowering-/.f64 N/A

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

      12. sub-neg N/A

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\color{blue}{\frac{2284450290879775841688574159837293}{625000000000000000000000000000} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}}{x}\right) \]

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

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\color{blue}{\frac{2284450290879775841688574159837293}{625000000000000000000000000000} \cdot \frac{1}{x} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}}{x}\right) \]

      14. associate-*r/ N/A

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\color{blue}{\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000} \cdot 1}{x}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}{x}\right) \]

      15. metadata-eval N/A

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\frac{\color{blue}{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}}{x} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}{x}\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto x \cdot \left(\frac{104109730557}{25000000000} + \frac{\color{blue}{\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)}{x}\right) \]

      17. metadata-eval 87.9

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

   5. Simplified 87.9%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7500000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 7:\\ \;\;\;\;\mathsf{fma}\left(-0.0424927283095952, z, x \cdot \mathsf{fma}\left(z, 0.28294182010212804, \mathsf{fma}\left(z, 0.0212463641547976, y \cdot -0.0424927283095952\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7500000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-78}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, y \cdot 0.3041881842569256, y \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 0.000155:\\ \;\;\;\;\frac{1}{\frac{-23.533438303}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7500000.0)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x -6.5e-78)
     (* x (fma x (* y 0.3041881842569256) (* y -0.0424927283095952)))
     (if (<= x 0.000155)
       (/ 1.0 (/ -23.533438303 z))
       (fma x 4.16438922228 -110.1139242984811)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7500000.0) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= -6.5e-78) {
		tmp = x * fma(x, (y * 0.3041881842569256), (y * -0.0424927283095952));
	} else if (x <= 0.000155) {
		tmp = 1.0 / (-23.533438303 / z);
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7500000.0)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= -6.5e-78)
		tmp = Float64(x * fma(x, Float64(y * 0.3041881842569256), Float64(y * -0.0424927283095952)));
	elseif (x <= 0.000155)
		tmp = Float64(1.0 / Float64(-23.533438303 / z));
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7500000.0], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, -6.5e-78], N[(x * N[(x * N[(y * 0.3041881842569256), $MachinePrecision] + N[(y * -0.0424927283095952), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000155], N[(1.0 / N[(-23.533438303 / z), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e6 or 1.55e-4 < x

   1. Initial program 17.4%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 14.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000} \cdot 1}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\color{blue}{\frac{104109730557}{25000000000}}}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000}}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot 1}{x \cdot z}}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

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

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\color{blue}{\frac{-13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\frac{-13764240537310136880149}{125000000000000000000}}{\color{blue}{z \cdot x}}\right)\right) \]

      13. *-lowering-*.f64 57.4

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

   8. Simplified 57.4%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x - \frac{13764240537310136880149}{125000000000000000000}} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{-13764240537310136880149}{125000000000000000000}} \]

       4. accelerator-lowering-fma.f64 88.6

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   11. Simplified 88.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   if -7.5e6 < x < -6.5000000000000003e-78

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 69.1

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

   5. Simplified 69.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y + x \cdot \left(\frac{500000000}{23533438303} \cdot y - \frac{-156699607947000000000}{553822718361107519809} \cdot y\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y + x \cdot \left(\frac{500000000}{23533438303} \cdot y - \frac{-156699607947000000000}{553822718361107519809} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{500000000}{23533438303} \cdot y - \frac{-156699607947000000000}{553822718361107519809} \cdot y\right) + \frac{-1000000000}{23533438303} \cdot y\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{500000000}{23533438303} \cdot y - \frac{-156699607947000000000}{553822718361107519809} \cdot y, \frac{-1000000000}{23533438303} \cdot y\right)} \]

      4. distribute-rgt-out-- N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{y \cdot \left(\frac{500000000}{23533438303} - \frac{-156699607947000000000}{553822718361107519809}\right)}, \frac{-1000000000}{23533438303} \cdot y\right) \]

      5. metadata-eval N/A

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

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

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

      7. *-lowering-*.f64 56.4

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

   8. Simplified 56.4%

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

   if -6.5000000000000003e-78 < x < 1.55e-4

   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. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. associate-/l/ N/A

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

      2. clear-num N/A

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

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

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

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

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{-23533438303}{1000000000}}{z}}} \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 67.1

         \[\leadsto \frac{1}{\color{blue}{\frac{-23.533438303}{z}}} \]

   9. Simplified 67.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{-23.533438303}{z}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7500000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-78}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, y \cdot 0.3041881842569256, y \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 0.000155:\\ \;\;\;\;\frac{1}{\frac{-23.533438303}{z}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7500000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 0.000155:\\ \;\;\;\;\mathsf{fma}\left(-0.0424927283095952, z, x \cdot \mathsf{fma}\left(z, 0.28294182010212804, \mathsf{fma}\left(z, 0.0212463641547976, y \cdot -0.0424927283095952\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7500000.0)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 0.000155)
     (fma
      -0.0424927283095952
      z
      (*
       x
       (fma
        z
        0.28294182010212804
        (fma z 0.0212463641547976 (* y -0.0424927283095952)))))
     (fma x 4.16438922228 -110.1139242984811))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7500000.0) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 0.000155) {
		tmp = fma(-0.0424927283095952, z, (x * fma(z, 0.28294182010212804, fma(z, 0.0212463641547976, (y * -0.0424927283095952)))));
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7500000.0)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 0.000155)
		tmp = fma(-0.0424927283095952, z, Float64(x * fma(z, 0.28294182010212804, fma(z, 0.0212463641547976, Float64(y * -0.0424927283095952)))));
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7500000.0], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 0.000155], N[(-0.0424927283095952 * z + N[(x * N[(z * 0.28294182010212804 + N[(z * 0.0212463641547976 + N[(y * -0.0424927283095952), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.5e6 or 1.55e-4 < x

   1. Initial program 17.4%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 14.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000} \cdot 1}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\color{blue}{\frac{104109730557}{25000000000}}}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000}}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot 1}{x \cdot z}}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

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

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\color{blue}{\frac{-13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\frac{-13764240537310136880149}{125000000000000000000}}{\color{blue}{z \cdot x}}\right)\right) \]

      13. *-lowering-*.f64 57.4

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

   8. Simplified 57.4%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x - \frac{13764240537310136880149}{125000000000000000000}} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{-13764240537310136880149}{125000000000000000000}} \]

       4. accelerator-lowering-fma.f64 88.6

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   11. Simplified 88.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   if -7.5e6 < x < 1.55e-4

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. flip3-- N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{{x}^{3} - {2}^{3}}{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}} \cdot \color{blue}{\frac{1}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

      5. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

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

         \[\leadsto \color{blue}{\frac{\frac{\left(\left(\left(x \cdot \frac{104109730557}{25000000000} + \frac{393497462077}{5000000000}\right) \cdot x + \frac{4297481763}{31250000}\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + \frac{216700011257}{5000000000}\right) \cdot x + \frac{263505074721}{1000000000}\right) \cdot x + \frac{156699607947}{500000000}\right) \cdot x + \frac{23533438303}{500000000}}}{\frac{x \cdot x + \left(2 \cdot 2 + x \cdot 2\right)}{{x}^{3} - {2}^{3}}}} \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}} \]
   5. Step-by-step derivation

      1. flip-+ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. sub-neg N/A

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

      4. swap-sqr N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. sub-neg N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. metadata-eval 99.6

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)} \]
   8. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right)} \]

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

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, \color{blue}{x \cdot \left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)}\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \color{blue}{\left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) + \left(\mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right) + \frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right)\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \left(\left(\mathsf{neg}\left(\color{blue}{z \cdot \frac{-156699607947000000000}{553822718361107519809}}\right)\right) + \frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \left(\color{blue}{z \cdot \left(\mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809}\right)\right)} + \frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right)\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \color{blue}{\mathsf{fma}\left(z, \mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809}\right), \frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right)\right)}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \mathsf{fma}\left(z, \color{blue}{\frac{156699607947000000000}{553822718361107519809}}, \frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right)\right)\right) \]

      9. distribute-lft-in N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \mathsf{fma}\left(z, \frac{156699607947000000000}{553822718361107519809}, \color{blue}{\frac{500000000}{23533438303} \cdot z + \frac{500000000}{23533438303} \cdot \left(-2 \cdot y\right)}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \mathsf{fma}\left(z, \frac{156699607947000000000}{553822718361107519809}, \color{blue}{z \cdot \frac{500000000}{23533438303}} + \frac{500000000}{23533438303} \cdot \left(-2 \cdot y\right)\right)\right) \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \mathsf{fma}\left(z, \frac{156699607947000000000}{553822718361107519809}, \color{blue}{\mathsf{fma}\left(z, \frac{500000000}{23533438303}, \frac{500000000}{23533438303} \cdot \left(-2 \cdot y\right)\right)}\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \mathsf{fma}\left(z, \frac{156699607947000000000}{553822718361107519809}, \mathsf{fma}\left(z, \frac{500000000}{23533438303}, \color{blue}{\left(\frac{500000000}{23533438303} \cdot -2\right) \cdot y}\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1000000000}{23533438303}, z, x \cdot \mathsf{fma}\left(z, \frac{156699607947000000000}{553822718361107519809}, \mathsf{fma}\left(z, \frac{500000000}{23533438303}, \color{blue}{\frac{-1000000000}{23533438303}} \cdot y\right)\right)\right) \]

      14. *-lowering-*.f64 93.4

          \[\leadsto \mathsf{fma}\left(-0.0424927283095952, z, x \cdot \mathsf{fma}\left(z, 0.28294182010212804, \mathsf{fma}\left(z, 0.0212463641547976, \color{blue}{-0.0424927283095952 \cdot y}\right)\right)\right) \]

   9. Simplified 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.0424927283095952, z, x \cdot \mathsf{fma}\left(z, 0.28294182010212804, \mathsf{fma}\left(z, 0.0212463641547976, -0.0424927283095952 \cdot y\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7500000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 0.000155:\\ \;\;\;\;\mathsf{fma}\left(-0.0424927283095952, z, x \cdot \mathsf{fma}\left(z, 0.28294182010212804, \mathsf{fma}\left(z, 0.0212463641547976, y \cdot -0.0424927283095952\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7500000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, y \cdot 0.3041881842569256, y \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 0.000155:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7500000.0)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x -4.4e-79)
     (* x (fma x (* y 0.3041881842569256) (* y -0.0424927283095952)))
     (if (<= x 0.000155)
       (* z -0.0424927283095952)
       (fma x 4.16438922228 -110.1139242984811)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7500000.0) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= -4.4e-79) {
		tmp = x * fma(x, (y * 0.3041881842569256), (y * -0.0424927283095952));
	} else if (x <= 0.000155) {
		tmp = z * -0.0424927283095952;
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7500000.0)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= -4.4e-79)
		tmp = Float64(x * fma(x, Float64(y * 0.3041881842569256), Float64(y * -0.0424927283095952)));
	elseif (x <= 0.000155)
		tmp = Float64(z * -0.0424927283095952);
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7500000.0], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, -4.4e-79], N[(x * N[(x * N[(y * 0.3041881842569256), $MachinePrecision] + N[(y * -0.0424927283095952), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000155], N[(z * -0.0424927283095952), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e6 or 1.55e-4 < x

   1. Initial program 17.4%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 14.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000} \cdot 1}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\color{blue}{\frac{104109730557}{25000000000}}}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000}}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot 1}{x \cdot z}}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

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

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\color{blue}{\frac{-13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\frac{-13764240537310136880149}{125000000000000000000}}{\color{blue}{z \cdot x}}\right)\right) \]

      13. *-lowering-*.f64 57.4

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

   8. Simplified 57.4%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x - \frac{13764240537310136880149}{125000000000000000000}} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{-13764240537310136880149}{125000000000000000000}} \]

       4. accelerator-lowering-fma.f64 88.6

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   11. Simplified 88.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   if -7.5e6 < x < -4.3999999999999998e-79

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

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

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

      10. +-commutative N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 69.1

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

   5. Simplified 69.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y + x \cdot \left(\frac{500000000}{23533438303} \cdot y - \frac{-156699607947000000000}{553822718361107519809} \cdot y\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y + x \cdot \left(\frac{500000000}{23533438303} \cdot y - \frac{-156699607947000000000}{553822718361107519809} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\frac{500000000}{23533438303} \cdot y - \frac{-156699607947000000000}{553822718361107519809} \cdot y\right) + \frac{-1000000000}{23533438303} \cdot y\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{500000000}{23533438303} \cdot y - \frac{-156699607947000000000}{553822718361107519809} \cdot y, \frac{-1000000000}{23533438303} \cdot y\right)} \]

      4. distribute-rgt-out-- N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{y \cdot \left(\frac{500000000}{23533438303} - \frac{-156699607947000000000}{553822718361107519809}\right)}, \frac{-1000000000}{23533438303} \cdot y\right) \]

      5. metadata-eval N/A

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

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

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

      7. *-lowering-*.f64 56.4

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

   8. Simplified 56.4%

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

   if -4.3999999999999998e-79 < x < 1.55e-4

   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

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot \frac{-1000000000}{23533438303}} \]

      2. *-lowering-*.f64 67.1

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

   5. Simplified 67.1%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7500000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq -4.4 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, y \cdot 0.3041881842569256, y \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 0.000155:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 76.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7500000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq -7.8 \cdot 10^{-78}:\\ \;\;\;\;x \cdot \left(y \cdot \mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right)\right)\\ \mathbf{elif}\;x \leq 0.000155:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7500000.0)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x -7.8e-78)
     (* x (* y (fma x 0.3041881842569256 -0.0424927283095952)))
     (if (<= x 0.000155)
       (* z -0.0424927283095952)
       (fma x 4.16438922228 -110.1139242984811)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7500000.0) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= -7.8e-78) {
		tmp = x * (y * fma(x, 0.3041881842569256, -0.0424927283095952));
	} else if (x <= 0.000155) {
		tmp = z * -0.0424927283095952;
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7500000.0)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= -7.8e-78)
		tmp = Float64(x * Float64(y * fma(x, 0.3041881842569256, -0.0424927283095952)));
	elseif (x <= 0.000155)
		tmp = Float64(z * -0.0424927283095952);
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7500000.0], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, -7.8e-78], N[(x * N[(y * N[(x * 0.3041881842569256 + -0.0424927283095952), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.000155], N[(z * -0.0424927283095952), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.5e6 or 1.55e-4 < x

   1. Initial program 17.4%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 14.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000} \cdot 1}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\color{blue}{\frac{104109730557}{25000000000}}}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000}}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot 1}{x \cdot z}}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

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

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\color{blue}{\frac{-13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\frac{-13764240537310136880149}{125000000000000000000}}{\color{blue}{z \cdot x}}\right)\right) \]

      13. *-lowering-*.f64 57.4

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

   8. Simplified 57.4%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x - \frac{13764240537310136880149}{125000000000000000000}} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{-13764240537310136880149}{125000000000000000000}} \]

       4. accelerator-lowering-fma.f64 88.6

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   11. Simplified 88.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   if -7.5e6 < x < -7.8000000000000004e-78

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

   5. Simplified 75.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y + x \cdot \left(\frac{500000000}{23533438303} \cdot \left(y - \frac{4297481763}{15625000}\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot y\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y + x \cdot \left(\frac{500000000}{23533438303} \cdot \left(y - \frac{4297481763}{15625000}\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot y\right)\right)} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, x \cdot \left(\frac{500000000}{23533438303} \cdot \left(y - \frac{4297481763}{15625000}\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot y\right)\right)} \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, \color{blue}{x \cdot \left(\frac{500000000}{23533438303} \cdot \left(y - \frac{4297481763}{15625000}\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot y\right)}\right) \]

      4. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, x \cdot \color{blue}{\left(\frac{500000000}{23533438303} \cdot \left(y - \frac{4297481763}{15625000}\right) + \left(\mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809} \cdot y\right)\right)\right)}\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, x \cdot \color{blue}{\mathsf{fma}\left(\frac{500000000}{23533438303}, y - \frac{4297481763}{15625000}, \mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809} \cdot y\right)\right)}\right) \]

      6. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, x \cdot \mathsf{fma}\left(\frac{500000000}{23533438303}, \color{blue}{y + \left(\mathsf{neg}\left(\frac{4297481763}{15625000}\right)\right)}, \mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809} \cdot y\right)\right)\right) \]

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

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, x \cdot \mathsf{fma}\left(\frac{500000000}{23533438303}, \color{blue}{y + \left(\mathsf{neg}\left(\frac{4297481763}{15625000}\right)\right)}, \mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809} \cdot y\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, x \cdot \mathsf{fma}\left(\frac{500000000}{23533438303}, y + \color{blue}{\frac{-4297481763}{15625000}}, \mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809} \cdot y\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, x \cdot \mathsf{fma}\left(\frac{500000000}{23533438303}, y + \frac{-4297481763}{15625000}, \mathsf{neg}\left(\color{blue}{y \cdot \frac{-156699607947000000000}{553822718361107519809}}\right)\right)\right) \]

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

          \[\leadsto x \cdot \mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, x \cdot \mathsf{fma}\left(\frac{500000000}{23533438303}, y + \frac{-4297481763}{15625000}, \color{blue}{y \cdot \left(\mathsf{neg}\left(\frac{-156699607947000000000}{553822718361107519809}\right)\right)}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto x \cdot \mathsf{fma}\left(\frac{-1000000000}{23533438303}, y, x \cdot \mathsf{fma}\left(\frac{500000000}{23533438303}, y + \frac{-4297481763}{15625000}, y \cdot \color{blue}{\frac{156699607947000000000}{553822718361107519809}}\right)\right) \]

      12. *-lowering-*.f64 62.7

          \[\leadsto x \cdot \mathsf{fma}\left(-0.0424927283095952, y, x \cdot \mathsf{fma}\left(0.0212463641547976, y + -275.038832832, \color{blue}{y \cdot 0.28294182010212804}\right)\right) \]

   8. Simplified 62.7%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(-0.0424927283095952, y, x \cdot \mathsf{fma}\left(0.0212463641547976, y + -275.038832832, y \cdot 0.28294182010212804\right)\right)} \]

   9. Taylor expanded in y around inf

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

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

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

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

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

       3. sub-neg N/A

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

       4. *-commutative N/A

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

       5. metadata-eval N/A

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

       6. accelerator-lowering-fma.f64 56.4

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

   11. Simplified 56.4%

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

   if -7.8000000000000004e-78 < x < 1.55e-4

   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

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot \frac{-1000000000}{23533438303}} \]

      2. *-lowering-*.f64 67.1

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

   5. Simplified 67.1%

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

Alternative 17: 76.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 0.000155:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.5)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 0.000155)
     (* z -0.0424927283095952)
     (fma x 4.16438922228 -110.1139242984811))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.5) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 0.000155) {
		tmp = z * -0.0424927283095952;
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.5)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 0.000155)
		tmp = Float64(z * -0.0424927283095952);
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.5], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 0.000155], N[(z * -0.0424927283095952), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5 or 1.55e-4 < x

   1. Initial program 18.0%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 14.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

      4. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000} \cdot 1}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\color{blue}{\frac{104109730557}{25000000000}}}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

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

         \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000}}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot 1}{x \cdot z}}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right)\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

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

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\color{blue}{\frac{-13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right) \]

      12. *-commutative N/A

          \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\frac{-13764240537310136880149}{125000000000000000000}}{\color{blue}{z \cdot x}}\right)\right) \]

      13. *-lowering-*.f64 57.0

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

   8. Simplified 57.0%

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

   9. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x - \frac{13764240537310136880149}{125000000000000000000}} \]
   10. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \color{blue}{\frac{104109730557}{25000000000} \cdot x + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{-13764240537310136880149}{125000000000000000000}} \]

       4. accelerator-lowering-fma.f64 88.0

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   11. Simplified 88.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]

   if -5.5 < x < 1.55e-4

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot \frac{-1000000000}{23533438303}} \]

      2. *-lowering-*.f64 62.6

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

   5. Simplified 62.6%

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

Alternative 18: 76.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.039:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.039)
   (* x 4.16438922228)
   (if (<= x 2.0) (* z -0.0424927283095952) (* x 4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.039) {
		tmp = x * 4.16438922228;
	} else if (x <= 2.0) {
		tmp = z * -0.0424927283095952;
	} else {
		tmp = x * 4.16438922228;
	}
	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 <= (-0.039d0)) then
        tmp = x * 4.16438922228d0
    else if (x <= 2.0d0) then
        tmp = z * (-0.0424927283095952d0)
    else
        tmp = x * 4.16438922228d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.039) {
		tmp = x * 4.16438922228;
	} else if (x <= 2.0) {
		tmp = z * -0.0424927283095952;
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.039:
		tmp = x * 4.16438922228
	elif x <= 2.0:
		tmp = z * -0.0424927283095952
	else:
		tmp = x * 4.16438922228
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.039)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= 2.0)
		tmp = Float64(z * -0.0424927283095952);
	else
		tmp = Float64(x * 4.16438922228);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.039)
		tmp = x * 4.16438922228;
	elseif (x <= 2.0)
		tmp = z * -0.0424927283095952;
	else
		tmp = x * 4.16438922228;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.039], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, 2.0], N[(z * -0.0424927283095952), $MachinePrecision], N[(x * 4.16438922228), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0389999999999999999 or 2 < x

   1. Initial program 17.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000}} \]

      2. *-lowering-*.f64 87.7

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

   5. Simplified 87.7%

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

   if -0.0389999999999999999 < x < 2

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot \frac{-1000000000}{23533438303}} \]

      2. *-lowering-*.f64 62.1

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

   5. Simplified 62.1%

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

Alternative 19: 45.2% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ x \cdot 4.16438922228 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x 4.16438922228))

C

double code(double x, double y, double z) {
	return x * 4.16438922228;
}

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 * 4.16438922228d0
end function

Java

public static double code(double x, double y, double z) {
	return x * 4.16438922228;
}

Python

def code(x, y, z):
	return x * 4.16438922228

Julia

function code(x, y, z)
	return Float64(x * 4.16438922228)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * 4.16438922228;
end

Wolfram

code[x_, y_, z_] := N[(x * 4.16438922228), $MachinePrecision]

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000}} \]

   2. *-lowering-*.f64 45.5

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

5. Simplified 45.5%

   \[\leadsto \color{blue}{x \cdot 4.16438922228} \]
6. Add Preprocessing

Alternative 20: 3.4% accurate, 79.0× speedup

Math

\[\begin{array}{l} \\ -110.1139242984811 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -110.1139242984811)

C

double code(double x, double y, double z) {
	return -110.1139242984811;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -110.1139242984811d0
end function

Java

public static double code(double x, double y, double z) {
	return -110.1139242984811;
}

Python

def code(x, y, z):
	return -110.1139242984811

Julia

function code(x, y, z)
	return -110.1139242984811
end

MATLAB

function tmp = code(x, y, z)
	tmp = -110.1139242984811;
end

Wolfram

code[x_, y_, z_] := -110.1139242984811

Derivation

1. Initial program 58.5%

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

3. Taylor expanded in z around inf

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

5. Simplified 55.0%

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

6. Taylor expanded in x around inf

   \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]
7. Step-by-step derivation

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

      \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

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

      \[\leadsto z \cdot \left(x \cdot \color{blue}{\left(\frac{104109730557}{25000000000} \cdot \frac{1}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)}\right) \]

   4. associate-*r/ N/A

      \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000} \cdot 1}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto z \cdot \left(x \cdot \left(\frac{\color{blue}{\frac{104109730557}{25000000000}}}{z} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

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

      \[\leadsto z \cdot \left(x \cdot \left(\color{blue}{\frac{\frac{104109730557}{25000000000}}{z}} + \left(\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x \cdot z}\right)\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot 1}{x \cdot z}}\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right)\right)\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

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

       \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \color{blue}{\frac{\mathsf{neg}\left(\frac{13764240537310136880149}{125000000000000000000}\right)}{x \cdot z}}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\color{blue}{\frac{-13764240537310136880149}{125000000000000000000}}}{x \cdot z}\right)\right) \]

   12. *-commutative N/A

       \[\leadsto z \cdot \left(x \cdot \left(\frac{\frac{104109730557}{25000000000}}{z} + \frac{\frac{-13764240537310136880149}{125000000000000000000}}{\color{blue}{z \cdot x}}\right)\right) \]

   13. *-lowering-*.f64 30.2

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

8. Simplified 30.2%

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

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-13764240537310136880149}{125000000000000000000}} \]
10. Step-by-step derivation

11. Simplified 3.3%

    \[\leadsto \color{blue}{-110.1139242984811} \]
12. Add Preprocessing

Developer Target 1: 98.8% 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 2024204 
(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)))