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

Percentage Accurate: 57.8% → 98.5%

Time: 18.5s

Alternatives: 18

Speedup: 4.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 57.8% 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: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right) \cdot \frac{1}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

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))
      1e+306)
   (*
    (*
     (fma x x -4.0)
     (/
      (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)))
    (/ 1.0 (+ x 2.0)))
   (fma x 4.16438922228 (/ y (* x 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)) <= 1e+306) {
		tmp = (fma(x, x, -4.0) * (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))) * (1.0 / (x + 2.0));
	} else {
		tmp = fma(x, 4.16438922228, (y / (x * 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)) <= 1e+306)
		tmp = Float64(Float64(fma(x, x, -4.0) * Float64(fma(x, fma(x, fma(x, 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 = fma(x, 4.16438922228, Float64(y / Float64(x * 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], 1e+306], N[(N[(N[(x * x + -4.0), $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $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]), $MachinePrecision] * N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $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))) < 1.00000000000000002e306

   1. Initial program 96.1%

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

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   4. Applied egg-rr 98.9%

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

   if 1.00000000000000002e306 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.2%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 99.2

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

   11. Simplified 99.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right) \cdot \frac{1}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

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))
      1e+306)
   (/
    (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)))
   (fma x 4.16438922228 (/ y (* x 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)) <= 1e+306) {
		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 = fma(x, 4.16438922228, (y / (x * 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)) <= 1e+306)
		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 = fma(x, 4.16438922228, Float64(y / Float64(x * 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], 1e+306], 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[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $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))) < 1.00000000000000002e306

   1. Initial program 96.1%

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

      1. *-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%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      8. /-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 1.00000000000000002e306 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.2%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 99.2

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

   11. Simplified 99.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\frac{x + -2}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

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))
      1e+306)
   (/
    (+ 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)))
   (fma x 4.16438922228 (/ y (* x 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)) <= 1e+306) {
		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 = fma(x, 4.16438922228, (y / (x * 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)) <= 1e+306)
		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 = fma(x, 4.16438922228, Float64(y / Float64(x * 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], 1e+306], 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[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $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))) < 1.00000000000000002e306

   1. Initial program 96.1%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   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. 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 98.8%

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

   if 1.00000000000000002e306 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.2%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 99.2

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

   11. Simplified 99.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\frac{x + -2}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

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))
      1e+306)
   (*
    (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)))
   (fma x 4.16438922228 (/ y (* x 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)) <= 1e+306) {
		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 = fma(x, 4.16438922228, (y / (x * 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)) <= 1e+306)
		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 = fma(x, 4.16438922228, Float64(y / Float64(x * 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], 1e+306], 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[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $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))) < 1.00000000000000002e306

   1. Initial program 96.1%

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

      1. *-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 1.00000000000000002e306 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.2%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 99.2

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

   11. Simplified 99.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\frac{-110.1139242984811 + \frac{\left(\frac{y}{x} - -3655.1204654076414\right) - \frac{130977.50649958357}{x}}{x}}{x} - -4.16438922228\right)\\ \mathbf{if}\;x \leq -96000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 380000000000:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          x
          (-
           (/
            (+
             -110.1139242984811
             (/
              (- (- (/ y x) -3655.1204654076414) (/ 130977.50649958357 x))
              x))
            x)
           -4.16438922228))))
   (if (<= x -96000000000000.0)
     t_0
     (if (<= x 380000000000.0)
       (/
        (* (- 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 * (((-110.1139242984811 + ((((y / x) - -3655.1204654076414) - (130977.50649958357 / x)) / x)) / x) - -4.16438922228);
	double tmp;
	if (x <= -96000000000000.0) {
		tmp = t_0;
	} else if (x <= 380000000000.0) {
		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(-110.1139242984811 + Float64(Float64(Float64(Float64(y / x) - -3655.1204654076414) - Float64(130977.50649958357 / x)) / x)) / x) - -4.16438922228))
	tmp = 0.0
	if (x <= -96000000000000.0)
		tmp = t_0;
	elseif (x <= 380000000000.0)
		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[(-110.1139242984811 + N[(N[(N[(N[(y / x), $MachinePrecision] - -3655.1204654076414), $MachinePrecision] - N[(130977.50649958357 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -96000000000000.0], t$95$0, If[LessEqual[x, 380000000000.0], 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 < -9.6e13 or 3.8e11 < x

   1. Initial program 8.3%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 98.5%

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

   if -9.6e13 < x < 3.8e11

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. 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 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -96000000000000:\\ \;\;\;\;x \cdot \left(\frac{-110.1139242984811 + \frac{\left(\frac{y}{x} - -3655.1204654076414\right) - \frac{130977.50649958357}{x}}{x}}{x} - -4.16438922228\right)\\ \mathbf{elif}\;x \leq 380000000000:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-110.1139242984811 + \frac{\left(\frac{y}{x} - -3655.1204654076414\right) - \frac{130977.50649958357}{x}}{x}}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot x}, y, x \cdot 4.16438922228\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.1e+14)
   (fma (/ 1.0 (* x x)) y (* x 4.16438922228))
   (if (<= x 5.8e+15)
     (/
      (* (- x 2.0) (fma x (fma x 137.519416416 y) z))
      (+
       (*
        x
        (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
       47.066876606))
     (fma x 4.16438922228 (/ y (* x x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.1e+14) {
		tmp = fma((1.0 / (x * x)), y, (x * 4.16438922228));
	} else if (x <= 5.8e+15) {
		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 = fma(x, 4.16438922228, (y / (x * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.1e+14)
		tmp = fma(Float64(1.0 / Float64(x * x)), y, Float64(x * 4.16438922228));
	elseif (x <= 5.8e+15)
		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 = fma(x, 4.16438922228, Float64(y / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.1e+14], N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * y + N[(x * 4.16438922228), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e+15], 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], N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1e14

   1. Initial program 13.6%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 97.6

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

   8. Simplified 97.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 97.6

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

   11. Simplified 97.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{x \cdot x} + x \cdot \frac{104109730557}{25000000000}} \]

       2. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x}{y}}} + x \cdot \frac{104109730557}{25000000000} \]

       3. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x \cdot x} \cdot y} + x \cdot \frac{104109730557}{25000000000} \]

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

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

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

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

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

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

       7. *-lowering-*.f64 97.6

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

   13. Applied egg-rr 97.6%

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

   if -5.1e14 < x < 5.8e15

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. 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%

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

   if 5.8e15 < x

   1. Initial program 3.1%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 99.2

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

   11. Simplified 99.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.1 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot x}, y, x \cdot 4.16438922228\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot x}, y, x \cdot 4.16438922228\right)\\ \mathbf{elif}\;x \leq 360000000:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 10.238818846568002, -1.787568985856513\right), 0.3041881842569256\right), -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.175)
   (fma (/ 1.0 (* x x)) y (* x 4.16438922228))
   (if (<= x 360000000.0)
     (*
      (fma
       x
       (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
       z)
      (fma
       x
       (fma x (fma x 10.238818846568002 -1.787568985856513) 0.3041881842569256)
       -0.0424927283095952))
     (fma x 4.16438922228 (/ y (* x x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.175) {
		tmp = fma((1.0 / (x * x)), y, (x * 4.16438922228));
	} else if (x <= 360000000.0) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, fma(x, fma(x, 10.238818846568002, -1.787568985856513), 0.3041881842569256), -0.0424927283095952);
	} else {
		tmp = fma(x, 4.16438922228, (y / (x * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.175)
		tmp = fma(Float64(1.0 / Float64(x * x)), y, Float64(x * 4.16438922228));
	elseif (x <= 360000000.0)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, fma(x, fma(x, 10.238818846568002, -1.787568985856513), 0.3041881842569256), -0.0424927283095952));
	else
		tmp = fma(x, 4.16438922228, Float64(y / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.175], N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * y + N[(x * 4.16438922228), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 360000000.0], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * N[(x * N[(x * N[(x * 10.238818846568002 + -1.787568985856513), $MachinePrecision] + 0.3041881842569256), $MachinePrecision] + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.17499999999999999

   1. Initial program 13.6%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 97.6

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

   8. Simplified 97.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 97.6

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

   11. Simplified 97.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{x \cdot x} + x \cdot \frac{104109730557}{25000000000}} \]

       2. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x}{y}}} + x \cdot \frac{104109730557}{25000000000} \]

       3. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x \cdot x} \cdot y} + x \cdot \frac{104109730557}{25000000000} \]

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

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

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

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

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

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

       7. *-lowering-*.f64 97.6

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

   13. Applied egg-rr 97.6%

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

   if -0.17499999999999999 < x < 3.6e8

   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. *-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 99.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \color{blue}{\left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + x \cdot \left(\frac{3140446455626174059100348970313144550000000}{306719603372886620352117082586607327396481} \cdot x - \frac{23298017199368982832548000000000}{13033352773350869092174451844127}\right)\right) - \frac{1000000000}{23533438303}\right)} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \color{blue}{\left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + x \cdot \left(\frac{3140446455626174059100348970313144550000000}{306719603372886620352117082586607327396481} \cdot x - \frac{23298017199368982832548000000000}{13033352773350869092174451844127}\right)\right) + \left(\mathsf{neg}\left(\frac{1000000000}{23533438303}\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + x \cdot \left(\frac{3140446455626174059100348970313144550000000}{306719603372886620352117082586607327396481} \cdot x - \frac{23298017199368982832548000000000}{13033352773350869092174451844127}\right)\right) + \color{blue}{\frac{-1000000000}{23533438303}}\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{168466327098500000000}{553822718361107519809} + x \cdot \left(\frac{3140446455626174059100348970313144550000000}{306719603372886620352117082586607327396481} \cdot x - \frac{23298017199368982832548000000000}{13033352773350869092174451844127}\right), \frac{-1000000000}{23533438303}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{3140446455626174059100348970313144550000000}{306719603372886620352117082586607327396481} \cdot x - \frac{23298017199368982832548000000000}{13033352773350869092174451844127}\right) + \frac{168466327098500000000}{553822718361107519809}}, \frac{-1000000000}{23533438303}\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{3140446455626174059100348970313144550000000}{306719603372886620352117082586607327396481} \cdot x - \frac{23298017199368982832548000000000}{13033352773350869092174451844127}, \frac{168466327098500000000}{553822718361107519809}\right)}, \frac{-1000000000}{23533438303}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\frac{3140446455626174059100348970313144550000000}{306719603372886620352117082586607327396481} \cdot x + \left(\mathsf{neg}\left(\frac{23298017199368982832548000000000}{13033352773350869092174451844127}\right)\right)}, \frac{168466327098500000000}{553822718361107519809}\right), \frac{-1000000000}{23533438303}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{3140446455626174059100348970313144550000000}{306719603372886620352117082586607327396481}} + \left(\mathsf{neg}\left(\frac{23298017199368982832548000000000}{13033352773350869092174451844127}\right)\right), \frac{168466327098500000000}{553822718361107519809}\right), \frac{-1000000000}{23533438303}\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \frac{3140446455626174059100348970313144550000000}{306719603372886620352117082586607327396481} + \color{blue}{\frac{-23298017199368982832548000000000}{13033352773350869092174451844127}}, \frac{168466327098500000000}{553822718361107519809}\right), \frac{-1000000000}{23533438303}\right) \]

      9. accelerator-lowering-fma.f64 97.6

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 10.238818846568002, -1.787568985856513\right)}, 0.3041881842569256\right), -0.0424927283095952\right) \]

   7. Simplified 97.6%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 10.238818846568002, -1.787568985856513\right), 0.3041881842569256\right), -0.0424927283095952\right)} \]

   if 3.6e8 < x

   1. Initial program 4.5%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 98.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 98.6

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

   8. Simplified 98.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 98.6

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

   11. Simplified 98.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 95.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot x}, y, x \cdot 4.16438922228\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.787568985856513, 0.3041881842569256\right), -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.175)
   (fma (/ 1.0 (* x x)) y (* x 4.16438922228))
   (if (<= x 2.0)
     (*
      (fma
       x
       (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
       z)
      (fma
       x
       (fma x -1.787568985856513 0.3041881842569256)
       -0.0424927283095952))
     (fma x 4.16438922228 (/ y (* x x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.175) {
		tmp = fma((1.0 / (x * x)), y, (x * 4.16438922228));
	} else if (x <= 2.0) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, fma(x, -1.787568985856513, 0.3041881842569256), -0.0424927283095952);
	} else {
		tmp = fma(x, 4.16438922228, (y / (x * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.175)
		tmp = fma(Float64(1.0 / Float64(x * x)), y, Float64(x * 4.16438922228));
	elseif (x <= 2.0)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, fma(x, -1.787568985856513, 0.3041881842569256), -0.0424927283095952));
	else
		tmp = fma(x, 4.16438922228, Float64(y / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.175], N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * y + N[(x * 4.16438922228), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.0], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * N[(x * N[(x * -1.787568985856513 + 0.3041881842569256), $MachinePrecision] + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.17499999999999999

   1. Initial program 13.6%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 97.6

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

   8. Simplified 97.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 97.6

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

   11. Simplified 97.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{x \cdot x} + x \cdot \frac{104109730557}{25000000000}} \]

       2. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x}{y}}} + x \cdot \frac{104109730557}{25000000000} \]

       3. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x \cdot x} \cdot y} + x \cdot \frac{104109730557}{25000000000} \]

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

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

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

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

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

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

       7. *-lowering-*.f64 97.6

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

   13. Applied egg-rr 97.6%

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

   if -0.17499999999999999 < x < 2

   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. *-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 99.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \color{blue}{\left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x\right) - \frac{1000000000}{23533438303}\right)} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \color{blue}{\left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1000000000}{23533438303}\right)\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x\right) + \color{blue}{\frac{-1000000000}{23533438303}}\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x, \frac{-1000000000}{23533438303}\right)} \]

      4. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \color{blue}{\frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x + \frac{168466327098500000000}{553822718361107519809}}, \frac{-1000000000}{23533438303}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-23298017199368982832548000000000}{13033352773350869092174451844127}} + \frac{168466327098500000000}{553822718361107519809}, \frac{-1000000000}{23533438303}\right) \]

      6. accelerator-lowering-fma.f64 98.9

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, -1.787568985856513, 0.3041881842569256\right)}, -0.0424927283095952\right) \]

   7. Simplified 98.9%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.787568985856513, 0.3041881842569256\right), -0.0424927283095952\right)} \]

   if 2 < x

   1. Initial program 7.0%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 95.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 96.0

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

   8. Simplified 96.0%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 96.0

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

   11. Simplified 96.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 95.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot x}, y, x \cdot 4.16438922228\right)\\ \mathbf{elif}\;x \leq 360000000:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.175)
   (fma (/ 1.0 (* x x)) y (* x 4.16438922228))
   (if (<= x 360000000.0)
     (*
      (fma
       x
       (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
       z)
      (fma x 0.3041881842569256 -0.0424927283095952))
     (fma x 4.16438922228 (/ y (* x x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.175) {
		tmp = fma((1.0 / (x * x)), y, (x * 4.16438922228));
	} else if (x <= 360000000.0) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, 0.3041881842569256, -0.0424927283095952);
	} else {
		tmp = fma(x, 4.16438922228, (y / (x * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.175)
		tmp = fma(Float64(1.0 / Float64(x * x)), y, Float64(x * 4.16438922228));
	elseif (x <= 360000000.0)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, 0.3041881842569256, -0.0424927283095952));
	else
		tmp = fma(x, 4.16438922228, Float64(y / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.175], N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * y + N[(x * 4.16438922228), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 360000000.0], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * N[(x * 0.3041881842569256 + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.17499999999999999

   1. Initial program 13.6%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 97.6

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

   8. Simplified 97.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 97.6

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

   11. Simplified 97.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{x \cdot x} + x \cdot \frac{104109730557}{25000000000}} \]

       2. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x}{y}}} + x \cdot \frac{104109730557}{25000000000} \]

       3. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x \cdot x} \cdot y} + x \cdot \frac{104109730557}{25000000000} \]

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

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

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

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

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

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

       7. *-lowering-*.f64 97.6

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

   13. Applied egg-rr 97.6%

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

   if -0.17499999999999999 < x < 3.6e8

   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. *-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 99.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \color{blue}{\left(\frac{168466327098500000000}{553822718361107519809} \cdot x - \frac{1000000000}{23533438303}\right)} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \color{blue}{\left(\frac{168466327098500000000}{553822718361107519809} \cdot x + \left(\mathsf{neg}\left(\frac{1000000000}{23533438303}\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \left(\color{blue}{x \cdot \frac{168466327098500000000}{553822718361107519809}} + \left(\mathsf{neg}\left(\frac{1000000000}{23533438303}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \left(x \cdot \frac{168466327098500000000}{553822718361107519809} + \color{blue}{\frac{-1000000000}{23533438303}}\right) \]

      4. accelerator-lowering-fma.f64 96.9

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \color{blue}{\mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right)} \]

   7. Simplified 96.9%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \color{blue}{\mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right)} \]

   if 3.6e8 < x

   1. Initial program 4.5%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 98.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 98.6

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

   8. Simplified 98.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 98.6

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

   11. Simplified 98.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 94.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot x}, y, x \cdot 4.16438922228\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.175)
   (fma (/ 1.0 (* x x)) y (* x 4.16438922228))
   (if (<= x 2.0)
     (*
      (fma
       x
       (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
       z)
      -0.0424927283095952)
     (fma x 4.16438922228 (/ y (* x x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.175) {
		tmp = fma((1.0 / (x * x)), y, (x * 4.16438922228));
	} else if (x <= 2.0) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * -0.0424927283095952;
	} else {
		tmp = fma(x, 4.16438922228, (y / (x * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.175)
		tmp = fma(Float64(1.0 / Float64(x * x)), y, Float64(x * 4.16438922228));
	elseif (x <= 2.0)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * -0.0424927283095952);
	else
		tmp = fma(x, 4.16438922228, Float64(y / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.175], N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * y + N[(x * 4.16438922228), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.0], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * -0.0424927283095952), $MachinePrecision], N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.17499999999999999

   1. Initial program 13.6%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 97.6

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

   8. Simplified 97.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 97.6

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

   11. Simplified 97.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{x \cdot x} + x \cdot \frac{104109730557}{25000000000}} \]

       2. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x}{y}}} + x \cdot \frac{104109730557}{25000000000} \]

       3. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x \cdot x} \cdot y} + x \cdot \frac{104109730557}{25000000000} \]

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

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

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

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

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

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

       7. *-lowering-*.f64 97.6

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

   13. Applied egg-rr 97.6%

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

   if -0.17499999999999999 < x < 2

   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. *-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 99.4%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \color{blue}{\frac{-1000000000}{23533438303}} \]
   6. Step-by-step derivation

   7. Simplified 97.3%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \color{blue}{-0.0424927283095952} \]

   if 2 < x

   1. Initial program 7.0%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 95.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 96.0

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

   8. Simplified 96.0%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 96.0

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

   11. Simplified 96.0%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 92.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.14:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x \cdot x}, y, x \cdot 4.16438922228\right)\\ \mathbf{elif}\;x \leq 360000000:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, -0.0424927283095952, z \cdot 0.3041881842569256\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.14)
   (fma (/ 1.0 (* x x)) y (* x 4.16438922228))
   (if (<= x 360000000.0)
     (fma
      x
      (fma y -0.0424927283095952 (* z 0.3041881842569256))
      (* z -0.0424927283095952))
     (fma x 4.16438922228 (/ y (* x x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.14) {
		tmp = fma((1.0 / (x * x)), y, (x * 4.16438922228));
	} else if (x <= 360000000.0) {
		tmp = fma(x, fma(y, -0.0424927283095952, (z * 0.3041881842569256)), (z * -0.0424927283095952));
	} else {
		tmp = fma(x, 4.16438922228, (y / (x * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.14)
		tmp = fma(Float64(1.0 / Float64(x * x)), y, Float64(x * 4.16438922228));
	elseif (x <= 360000000.0)
		tmp = fma(x, fma(y, -0.0424927283095952, Float64(z * 0.3041881842569256)), Float64(z * -0.0424927283095952));
	else
		tmp = fma(x, 4.16438922228, Float64(y / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.14], N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * y + N[(x * 4.16438922228), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 360000000.0], N[(x * N[(y * -0.0424927283095952 + N[(z * 0.3041881842569256), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.14000000000000001

   1. Initial program 13.6%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 97.6

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

   8. Simplified 97.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 97.6

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

   11. Simplified 97.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{\frac{y}{x \cdot x} + x \cdot \frac{104109730557}{25000000000}} \]

       2. clear-num N/A

          \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x}{y}}} + x \cdot \frac{104109730557}{25000000000} \]

       3. associate-/r/ N/A

          \[\leadsto \color{blue}{\frac{1}{x \cdot x} \cdot y} + x \cdot \frac{104109730557}{25000000000} \]

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

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

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

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

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

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

       7. *-lowering-*.f64 97.6

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

   13. Applied egg-rr 97.6%

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

   if -0.14000000000000001 < x < 3.6e8

   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. flip-- N/A

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

      3. frac-times N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 93.5

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

   7. Simplified 93.5%

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

   if 3.6e8 < x

   1. Initial program 4.5%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 98.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 98.6

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

   8. Simplified 98.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 98.6

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

   11. Simplified 98.6%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 92.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -0.19:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 360000000:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, -0.0424927283095952, z \cdot 0.3041881842569256\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x 4.16438922228 (/ y (* x x)))))
   (if (<= x -0.19)
     t_0
     (if (<= x 360000000.0)
       (fma
        x
        (fma y -0.0424927283095952 (* z 0.3041881842569256))
        (* z -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, 4.16438922228, (y / (x * x)));
	double tmp;
	if (x <= -0.19) {
		tmp = t_0;
	} else if (x <= 360000000.0) {
		tmp = fma(x, fma(y, -0.0424927283095952, (z * 0.3041881842569256)), (z * -0.0424927283095952));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, 4.16438922228, Float64(y / Float64(x * x)))
	tmp = 0.0
	if (x <= -0.19)
		tmp = t_0;
	elseif (x <= 360000000.0)
		tmp = fma(x, fma(y, -0.0424927283095952, Float64(z * 0.3041881842569256)), Float64(z * -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.19], t$95$0, If[LessEqual[x, 360000000.0], N[(x * N[(y * -0.0424927283095952 + N[(z * 0.3041881842569256), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.19 or 3.6e8 < x

   1. Initial program 8.9%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 98.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 98.1

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

   8. Simplified 98.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 98.1

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

   11. Simplified 98.1%

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

   if -0.19 < x < 3.6e8

   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. flip-- N/A

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

      3. frac-times N/A

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

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

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 93.5

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

   7. Simplified 93.5%

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

Alternative 13: 80.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -0.122:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 360000000:\\ \;\;\;\;\mathsf{fma}\left(x, z \cdot 0.3041881842569256, z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x 4.16438922228 (/ y (* x x)))))
   (if (<= x -0.122)
     t_0
     (if (<= x 360000000.0)
       (fma x (* z 0.3041881842569256) (* z -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, 4.16438922228, (y / (x * x)));
	double tmp;
	if (x <= -0.122) {
		tmp = t_0;
	} else if (x <= 360000000.0) {
		tmp = fma(x, (z * 0.3041881842569256), (z * -0.0424927283095952));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, 4.16438922228, Float64(y / Float64(x * x)))
	tmp = 0.0
	if (x <= -0.122)
		tmp = t_0;
	elseif (x <= 360000000.0)
		tmp = fma(x, Float64(z * 0.3041881842569256), Float64(z * -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.122], t$95$0, If[LessEqual[x, 360000000.0], N[(x * N[(z * 0.3041881842569256), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.122 or 3.6e8 < x

   1. Initial program 8.9%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 98.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \color{blue}{\frac{y}{{x}^{3}}}\right)\right) \]

      2. cube-mult N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot \left(x \cdot x\right)}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{{x}^{2}}}\right)\right) \]

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

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{\color{blue}{x \cdot {x}^{2}}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \left(\frac{-104109730557}{25000000000} - \frac{y}{x \cdot \color{blue}{\left(x \cdot x\right)}}\right)\right) \]

      6. *-lowering-*.f64 98.1

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

   8. Simplified 98.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} + \frac{y}{{x}^{3}}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000} + x \cdot \frac{y}{{x}^{3}}} \]

       2. associate-*r/ N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x \cdot y}{{x}^{3}}} \]

       3. cube-mult N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

       4. unpow2 N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \frac{x \cdot y}{x \cdot \color{blue}{{x}^{2}}} \]

       5. times-frac N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{x}{x} \cdot \frac{y}{{x}^{2}}} \]

       6. *-inverses N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{1} \cdot \frac{y}{{x}^{2}} \]

       7. *-lft-identity N/A

          \[\leadsto x \cdot \frac{104109730557}{25000000000} + \color{blue}{\frac{y}{{x}^{2}}} \]

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

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{{x}^{2}}\right)} \]

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

          \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \color{blue}{\frac{y}{{x}^{2}}}\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{y}{\color{blue}{x \cdot x}}\right) \]

       11. *-lowering-*.f64 98.1

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

   11. Simplified 98.1%

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

   if -0.122 < x < 3.6e8

   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 z around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 73.6

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

   5. Simplified 73.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 71.4

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

   8. Simplified 71.4%

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

Alternative 14: 77.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{if}\;x \leq -82:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4500000000:\\ \;\;\;\;\mathsf{fma}\left(x, z \cdot 0.3041881842569256, z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ 4.16438922228 (/ -110.1139242984811 x)))))
   (if (<= x -82.0)
     t_0
     (if (<= x 4500000000.0)
       (fma x (* z 0.3041881842569256) (* z -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.16438922228 + (-110.1139242984811 / x));
	double tmp;
	if (x <= -82.0) {
		tmp = t_0;
	} else if (x <= 4500000000.0) {
		tmp = fma(x, (z * 0.3041881842569256), (z * -0.0424927283095952));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)))
	tmp = 0.0
	if (x <= -82.0)
		tmp = t_0;
	elseif (x <= 4500000000.0)
		tmp = fma(x, Float64(z * 0.3041881842569256), Float64(z * -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -82.0], t$95$0, If[LessEqual[x, 4500000000.0], N[(x * N[(z * 0.3041881842569256), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -82 or 4.5e9 < x

   1. Initial program 8.3%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} + \frac{104109730557}{25000000000}\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} - \frac{104109730557}{25000000000}\right)\right)} \]

      5. neg-sub0 N/A

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

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

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

      7. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} - \frac{104109730557}{25000000000}\right)\right)} \]

      8. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right) + \frac{104109730557}{25000000000}\right)} \]

      9. neg-sub0 N/A

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

      10. +-commutative N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

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

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

      16. metadata-eval 94.2

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

   5. Simplified 94.2%

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

   if -82 < x < 4.5e9

   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 z around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 73.0

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

   5. Simplified 73.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 70.8

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

   8. Simplified 70.8%

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

Alternative 15: 77.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 4500000000:\\ \;\;\;\;\mathsf{fma}\left(x, z \cdot 0.3041881842569256, z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.175)
   (* x 4.16438922228)
   (if (<= x 4500000000.0)
     (fma x (* z 0.3041881842569256) (* z -0.0424927283095952))
     (* x 4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.175) {
		tmp = x * 4.16438922228;
	} else if (x <= 4500000000.0) {
		tmp = fma(x, (z * 0.3041881842569256), (z * -0.0424927283095952));
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.175)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= 4500000000.0)
		tmp = fma(x, Float64(z * 0.3041881842569256), Float64(z * -0.0424927283095952));
	else
		tmp = Float64(x * 4.16438922228);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.175], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, 4500000000.0], N[(x * N[(z * 0.3041881842569256), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.17499999999999999 or 4.5e9 < x

   1. Initial program 8.3%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   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 94.2

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

   5. Simplified 94.2%

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

   if -0.17499999999999999 < x < 4.5e9

   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 z around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 73.0

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

   5. Simplified 73.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

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

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

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

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 70.8

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

   8. Simplified 70.8%

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

Alternative 16: 77.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -98:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 4500000000:\\ \;\;\;\;z \cdot \mathsf{fma}\left(0.3041881842569256, x, -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -98.0)
   (* x 4.16438922228)
   (if (<= x 4500000000.0)
     (* z (fma 0.3041881842569256 x -0.0424927283095952))
     (* x 4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -98.0) {
		tmp = x * 4.16438922228;
	} else if (x <= 4500000000.0) {
		tmp = z * fma(0.3041881842569256, x, -0.0424927283095952);
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -98.0)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= 4500000000.0)
		tmp = Float64(z * fma(0.3041881842569256, x, -0.0424927283095952));
	else
		tmp = Float64(x * 4.16438922228);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -98.0], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, 4500000000.0], N[(z * N[(0.3041881842569256 * x + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -98 or 4.5e9 < x

   1. Initial program 8.3%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   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 94.2

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

   5. Simplified 94.2%

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

   if -98 < x < 4.5e9

   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 z around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 73.0

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

   5. Simplified 73.0%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. accelerator-lowering-fma.f64 71.5

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

   8. Simplified 71.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

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

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

       4. associate-*l* N/A

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

       5. *-commutative N/A

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

       6. distribute-lft-out N/A

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

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

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

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

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

       9. metadata-eval 70.8

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

   11. Simplified 70.8%

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

Alternative 17: 77.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.195:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.195)
   (* 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 <= -0.195) {
		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 <= (-0.195d0)) 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 <= -0.195) {
		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 <= -0.195:
		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 <= -0.195)
		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 <= -0.195)
		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, -0.195], 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 < -0.19500000000000001 or 2 < x

   1. Initial program 10.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000}} \]

      2. *-lowering-*.f64 92.2

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

   5. Simplified 92.2%

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

   if -0.19500000000000001 < x < 2

   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 71.9

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

   5. Simplified 71.9%

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

Alternative 18: 45.1% 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 50.8%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{104109730557}{25000000000}} \]

   2. *-lowering-*.f64 51.9

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

5. Simplified 51.9%

   \[\leadsto \color{blue}{x \cdot 4.16438922228} \]
6. Add Preprocessing

Developer Target 1: 98.6% 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 2024199 
(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)))