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

Percentage Accurate: 57.7% → 97.5%

Time: 35.5s

Alternatives: 17

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: 57.7% 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: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{\left(y + -33.31511377824\right) + -130944.19138580533}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \end{array} \end{array} \]

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+282)
   (/
    (*
     (fma x (* x x) -8.0)
     (/
      (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)))
    (fma x x (fma x 2.0 4.0)))
   (*
    x
    (-
     (/
      (+
       (/
        (-
         (/ (+ (+ y -33.31511377824) -130944.19138580533) x)
         -3655.1204654076414)
        x)
       -110.1139242984811)
      x)
     -4.16438922228))))

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+282) {
		tmp = (fma(x, (x * x), -8.0) * (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))) / fma(x, x, fma(x, 2.0, 4.0));
	} else {
		tmp = x * ((((((((y + -33.31511377824) + -130944.19138580533) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228);
	}
	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+282)
		tmp = Float64(Float64(fma(x, Float64(x * x), -8.0) * 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))) / fma(x, x, fma(x, 2.0, 4.0)));
	else
		tmp = Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + -33.31511377824) + -130944.19138580533) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228));
	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+282], N[(N[(N[(x * N[(x * x), $MachinePrecision] + -8.0), $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[(x * x + N[(x * 2.0 + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(N[(N[(N[(N[(N[(y + -33.31511377824), $MachinePrecision] + -130944.19138580533), $MachinePrecision] / x), $MachinePrecision] - -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + -110.1139242984811), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $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))) < 1.00000000000000003e282

   1. Initial program 95.7%

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

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   if 1.00000000000000003e282 < (/.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 0.1%

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

   4. Applied rewrites 1.1%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{{x}^{2} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{{x}^{2} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)} \]

      4. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} + \left(\mathsf{neg}\left(1\right)\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} + \color{blue}{-1}\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} + -1\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{2 + 4 \cdot \frac{1}{x}}{x}\right)\right)} + -1\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2 + 4 \cdot \frac{1}{x}}{\mathsf{neg}\left(x\right)}} + -1\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + 4 \cdot \frac{1}{x}}{\color{blue}{-1 \cdot x}} + -1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2 + 4 \cdot \frac{1}{x}}{-1 \cdot x}} + -1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{\color{blue}{2 + 4 \cdot \frac{1}{x}}}{-1 \cdot x} + -1\right)} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \color{blue}{\frac{4 \cdot 1}{x}}}{-1 \cdot x} + -1\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{\color{blue}{4}}{x}}{-1 \cdot x} + -1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \color{blue}{\frac{4}{x}}}{-1 \cdot x} + -1\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{4}{x}}{\color{blue}{\mathsf{neg}\left(x\right)}} + -1\right)} \]

      16. lower-neg.f64 1.1

          \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{4}{x}}{\color{blue}{-x}} + -1\right)} \]

   7. Applied rewrites 1.1%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{4}{x}}{-x} + -1\right)}} \]

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{104109730557}{3125000000} + -1 \cdot y\right) - \frac{409200598080641655637810418659684985388407301}{3125000000000000000000000000000000000000}}{x} - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]

   10. Applied rewrites 99.1%

       \[\leadsto \color{blue}{\left(-x\right) \cdot \left(\left(-\frac{\left(-\frac{\left(-\frac{\left(-33.31511377824 + y\right) + -130944.19138580533}{x}\right) + -3655.1204654076414}{x}\right) + -110.1139242984811}{x}\right) + -4.16438922228\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{\left(y + -33.31511377824\right) + -130944.19138580533}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+282}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{\left(y + -33.31511377824\right) + -130944.19138580533}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \end{array} \end{array} \]

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+282)
   (/
    (/
     (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)))
   (*
    x
    (-
     (/
      (+
       (/
        (-
         (/ (+ (+ y -33.31511377824) -130944.19138580533) x)
         -3655.1204654076414)
        x)
       -110.1139242984811)
      x)
     -4.16438922228))))

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+282) {
		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)) / (1.0 / (x + -2.0));
	} else {
		tmp = x * ((((((((y + -33.31511377824) + -130944.19138580533) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228);
	}
	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+282)
		tmp = Float64(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 = Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + -33.31511377824) + -130944.19138580533) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228));
	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+282], N[(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] / N[(1.0 / N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(N[(N[(N[(N[(N[(y + -33.31511377824), $MachinePrecision] + -130944.19138580533), $MachinePrecision] / x), $MachinePrecision] - -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + -110.1139242984811), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $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))) < 1.00000000000000003e282

   1. Initial program 95.7%

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

   4. Applied rewrites 98.8%

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

   if 1.00000000000000003e282 < (/.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 0.1%

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

   4. Applied rewrites 1.1%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{{x}^{2} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{{x}^{2} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)} \]

      4. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} + \left(\mathsf{neg}\left(1\right)\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} + \color{blue}{-1}\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} + -1\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{2 + 4 \cdot \frac{1}{x}}{x}\right)\right)} + -1\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2 + 4 \cdot \frac{1}{x}}{\mathsf{neg}\left(x\right)}} + -1\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + 4 \cdot \frac{1}{x}}{\color{blue}{-1 \cdot x}} + -1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2 + 4 \cdot \frac{1}{x}}{-1 \cdot x}} + -1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{\color{blue}{2 + 4 \cdot \frac{1}{x}}}{-1 \cdot x} + -1\right)} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \color{blue}{\frac{4 \cdot 1}{x}}}{-1 \cdot x} + -1\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{\color{blue}{4}}{x}}{-1 \cdot x} + -1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \color{blue}{\frac{4}{x}}}{-1 \cdot x} + -1\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{4}{x}}{\color{blue}{\mathsf{neg}\left(x\right)}} + -1\right)} \]

      16. lower-neg.f64 1.1

          \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{4}{x}}{\color{blue}{-x}} + -1\right)} \]

   7. Applied rewrites 1.1%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{4}{x}}{-x} + -1\right)}} \]

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{104109730557}{3125000000} + -1 \cdot y\right) - \frac{409200598080641655637810418659684985388407301}{3125000000000000000000000000000000000000}}{x} - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]

   10. Applied rewrites 99.1%

       \[\leadsto \color{blue}{\left(-x\right) \cdot \left(\left(-\frac{\left(-\frac{\left(-\frac{\left(-33.31511377824 + y\right) + -130944.19138580533}{x}\right) + -3655.1204654076414}{x}\right) + -110.1139242984811}{x}\right) + -4.16438922228\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+282}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{\left(y + -33.31511377824\right) + -130944.19138580533}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)} \cdot \left(x + -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{\left(y + -33.31511377824\right) + -130944.19138580533}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \end{array} \end{array} \]

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+282)
   (*
    (/
     (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))
   (*
    x
    (-
     (/
      (+
       (/
        (-
         (/ (+ (+ y -33.31511377824) -130944.19138580533) x)
         -3655.1204654076414)
        x)
       -110.1139242984811)
      x)
     -4.16438922228))))

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+282) {
		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 = x * ((((((((y + -33.31511377824) + -130944.19138580533) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228);
	}
	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+282)
		tmp = Float64(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(x + -2.0));
	else
		tmp = Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + -33.31511377824) + -130944.19138580533) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228));
	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+282], N[(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] * N[(x + -2.0), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(N[(N[(N[(N[(N[(y + -33.31511377824), $MachinePrecision] + -130944.19138580533), $MachinePrecision] / x), $MachinePrecision] - -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + -110.1139242984811), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $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))) < 1.00000000000000003e282

   1. Initial program 95.7%

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

   4. Applied rewrites 98.8%

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

   if 1.00000000000000003e282 < (/.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 0.1%

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

   4. Applied rewrites 1.1%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{{x}^{2} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{{x}^{2} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)} \]

      4. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} + \left(\mathsf{neg}\left(1\right)\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} + \color{blue}{-1}\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} + -1\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{2 + 4 \cdot \frac{1}{x}}{x}\right)\right)} + -1\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2 + 4 \cdot \frac{1}{x}}{\mathsf{neg}\left(x\right)}} + -1\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + 4 \cdot \frac{1}{x}}{\color{blue}{-1 \cdot x}} + -1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2 + 4 \cdot \frac{1}{x}}{-1 \cdot x}} + -1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{\color{blue}{2 + 4 \cdot \frac{1}{x}}}{-1 \cdot x} + -1\right)} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \color{blue}{\frac{4 \cdot 1}{x}}}{-1 \cdot x} + -1\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{\color{blue}{4}}{x}}{-1 \cdot x} + -1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \color{blue}{\frac{4}{x}}}{-1 \cdot x} + -1\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{4}{x}}{\color{blue}{\mathsf{neg}\left(x\right)}} + -1\right)} \]

      16. lower-neg.f64 1.1

          \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{4}{x}}{\color{blue}{-x}} + -1\right)} \]

   7. Applied rewrites 1.1%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{4}{x}}{-x} + -1\right)}} \]

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{104109730557}{3125000000} + -1 \cdot y\right) - \frac{409200598080641655637810418659684985388407301}{3125000000000000000000000000000000000000}}{x} - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]

   10. Applied rewrites 99.1%

       \[\leadsto \color{blue}{\left(-x\right) \cdot \left(\left(-\frac{\left(-\frac{\left(-\frac{\left(-33.31511377824 + y\right) + -130944.19138580533}{x}\right) + -3655.1204654076414}{x}\right) + -110.1139242984811}{x}\right) + -4.16438922228\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)} \cdot \left(x + -2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{\left(y + -33.31511377824\right) + -130944.19138580533}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\frac{\frac{\frac{\left(y + -33.31511377824\right) + -130944.19138580533}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \mathbf{if}\;x \leq -6500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9500000:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          x
          (-
           (/
            (+
             (/
              (-
               (/ (+ (+ y -33.31511377824) -130944.19138580533) x)
               -3655.1204654076414)
              x)
             -110.1139242984811)
            x)
           -4.16438922228))))
   (if (<= x -6500000000000.0)
     t_0
     (if (<= x 9500000.0)
       (/
        (* (- x 2.0) (fma x (fma x 137.519416416 y) z))
        (+
         (*
          x
          (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
         47.066876606))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * ((((((((y + -33.31511377824) + -130944.19138580533) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228);
	double tmp;
	if (x <= -6500000000000.0) {
		tmp = t_0;
	} else if (x <= 9500000.0) {
		tmp = ((x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + -33.31511377824) + -130944.19138580533) / x) - -3655.1204654076414) / x) + -110.1139242984811) / x) - -4.16438922228))
	tmp = 0.0
	if (x <= -6500000000000.0)
		tmp = t_0;
	elseif (x <= 9500000.0)
		tmp = Float64(Float64(Float64(x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[(N[(N[(N[(N[(N[(N[(y + -33.31511377824), $MachinePrecision] + -130944.19138580533), $MachinePrecision] / x), $MachinePrecision] - -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + -110.1139242984811), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6500000000000.0], t$95$0, If[LessEqual[x, 9500000.0], 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 < -6.5e12 or 9.5e6 < x

   1. Initial program 14.8%

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

   4. Applied rewrites 19.1%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{{x}^{2} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{{x}^{2} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\color{blue}{\left(x \cdot x\right)} \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} - 1\right)} \]

      4. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} + \left(\mathsf{neg}\left(1\right)\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} + \color{blue}{-1}\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \color{blue}{\left(-1 \cdot \frac{2 + 4 \cdot \frac{1}{x}}{x} + -1\right)}} \]

      7. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{2 + 4 \cdot \frac{1}{x}}{x}\right)\right)} + -1\right)} \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2 + 4 \cdot \frac{1}{x}}{\mathsf{neg}\left(x\right)}} + -1\right)} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + 4 \cdot \frac{1}{x}}{\color{blue}{-1 \cdot x}} + -1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\color{blue}{\frac{2 + 4 \cdot \frac{1}{x}}{-1 \cdot x}} + -1\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{\color{blue}{2 + 4 \cdot \frac{1}{x}}}{-1 \cdot x} + -1\right)} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \color{blue}{\frac{4 \cdot 1}{x}}}{-1 \cdot x} + -1\right)} \]

      13. metadata-eval N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{\color{blue}{4}}{x}}{-1 \cdot x} + -1\right)} \]

      14. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \color{blue}{\frac{4}{x}}}{-1 \cdot x} + -1\right)} \]

      15. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + \frac{216700011257}{5000000000}, \frac{263505074721}{1000000000}\right), \frac{156699607947}{500000000}\right), \frac{23533438303}{500000000}\right)}\right)}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{4}{x}}{\color{blue}{\mathsf{neg}\left(x\right)}} + -1\right)} \]

      16. lower-neg.f64 19.2

          \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{4}{x}}{\color{blue}{-x}} + -1\right)} \]

   7. Applied rewrites 19.2%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\color{blue}{\left(x \cdot x\right) \cdot \left(\frac{2 + \frac{4}{x}}{-x} + -1\right)}} \]

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \left(\frac{104109730557}{3125000000} + -1 \cdot y\right) - \frac{409200598080641655637810418659684985388407301}{3125000000000000000000000000000000000000}}{x} - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]

   10. Applied rewrites 95.5%

       \[\leadsto \color{blue}{\left(-x\right) \cdot \left(\left(-\frac{\left(-\frac{\left(-\frac{\left(-33.31511377824 + y\right) + -130944.19138580533}{x}\right) + -3655.1204654076414}{x}\right) + -110.1139242984811}{x}\right) + -4.16438922228\right)} \]

   if -6.5e12 < x < 9.5e6

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-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. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6500000000000:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{\left(y + -33.31511377824\right) + -130944.19138580533}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \mathbf{elif}\;x \leq 9500000:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{\left(y + -33.31511377824\right) + -130944.19138580533}{x} - -3655.1204654076414}{x} + -110.1139242984811}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -32000000000000:\\ \;\;\;\;\frac{x \cdot \left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right)}{4.16438922228 + \frac{110.1139242984811}{x}}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -32000000000000.0)
   (/
    (* x (- 17.342137594641823 (/ 12125.076324411626 (* x x))))
    (+ 4.16438922228 (/ 110.1139242984811 x)))
   (if (<= x 1.6e+35)
     (/
      (* (- x 2.0) (fma x (fma x 137.519416416 y) z))
      (+
       (*
        x
        (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
       47.066876606))
     (* x 4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -32000000000000.0) {
		tmp = (x * (17.342137594641823 - (12125.076324411626 / (x * x)))) / (4.16438922228 + (110.1139242984811 / x));
	} else if (x <= 1.6e+35) {
		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 = x * 4.16438922228;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -32000000000000.0)
		tmp = Float64(Float64(x * Float64(17.342137594641823 - Float64(12125.076324411626 / Float64(x * x)))) / Float64(4.16438922228 + Float64(110.1139242984811 / x)));
	elseif (x <= 1.6e+35)
		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 = Float64(x * 4.16438922228);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -32000000000000.0], N[(N[(x * N[(17.342137594641823 - N[(12125.076324411626 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.16438922228 + N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e+35], 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), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2e13

   1. Initial program 13.4%

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

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \left(\color{blue}{x \cdot \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}, \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

   7. Applied rewrites 22.9%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 90.0

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

   10. Applied rewrites 90.0%

       \[\leadsto \color{blue}{x \cdot \left(4.16438922228 - \frac{110.1139242984811}{x}\right)} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. *-commutative N/A

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

       4. lift--.f64 N/A

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

       5. flip-- N/A

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

       6. associate-*l/ N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. metadata-eval N/A

           \[\leadsto \frac{\left(\color{blue}{\frac{10838835996651139530249}{625000000000000000000}} - \frac{\frac{13764240537310136880149}{125000000000000000000}}{x} \cdot \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \cdot \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       12. lift-/.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \frac{\frac{13764240537310136880149}{125000000000000000000}}{x} \cdot \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000}}{x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       13. frac-times N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{13764240537310136880149}{125000000000000000000}}{x \cdot x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       14. lift-*.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \frac{\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{13764240537310136880149}{125000000000000000000}}{\color{blue}{x \cdot x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       15. lower-/.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{13764240537310136880149}{125000000000000000000}}{x \cdot x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       16. metadata-eval N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \frac{\color{blue}{\frac{189454317568931645604627337684454555190262201}{15625000000000000000000000000000000000000}}}{x \cdot x}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       17. lower-+.f64 90.3

           \[\leadsto \frac{\left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right) \cdot x}{\color{blue}{4.16438922228 + \frac{110.1139242984811}{x}}} \]

   12. Applied rewrites 90.3%

       \[\leadsto \color{blue}{\frac{\left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right) \cdot x}{4.16438922228 + \frac{110.1139242984811}{x}}} \]

   if -3.2e13 < x < 1.59999999999999991e35

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-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. lower-fma.f64 96.4

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

   5. Applied rewrites 96.4%

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

   if 1.59999999999999991e35 < x

   1. Initial program 5.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 94.3

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

   5. Applied rewrites 94.3%

      \[\leadsto \color{blue}{x \cdot 4.16438922228} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -32000000000000:\\ \;\;\;\;\frac{x \cdot \left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right)}{4.16438922228 + \frac{110.1139242984811}{x}}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-0.0424927283095952, y, z \cdot 0.3041881842569256\right)\\ \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\frac{x \cdot \left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right)}{4.16438922228 + \frac{110.1139242984811}{x}}\\ \mathbf{elif}\;x \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(z, -0.38999068429136097, \mathsf{fma}\left(t\_0, 7.158593866711955, 5.843575199059173\right)\right), -x, t\_0\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma -0.0424927283095952 y (* z 0.3041881842569256))))
   (if (<= x -43000000000.0)
     (/
      (* x (- 17.342137594641823 (/ 12125.076324411626 (* x x))))
      (+ 4.16438922228 (/ 110.1139242984811 x)))
     (if (<= x 0.4)
       (fma
        x
        (fma
         (fma
          z
          -0.38999068429136097
          (fma t_0 7.158593866711955 5.843575199059173))
         (- x)
         t_0)
        (* z -0.0424927283095952))
       (*
        x
        (+
         4.16438922228
         (/ (+ -110.1139242984811 (/ 3655.1204654076414 x)) x)))))))

C

double code(double x, double y, double z) {
	double t_0 = fma(-0.0424927283095952, y, (z * 0.3041881842569256));
	double tmp;
	if (x <= -43000000000.0) {
		tmp = (x * (17.342137594641823 - (12125.076324411626 / (x * x)))) / (4.16438922228 + (110.1139242984811 / x));
	} else if (x <= 0.4) {
		tmp = fma(x, fma(fma(z, -0.38999068429136097, fma(t_0, 7.158593866711955, 5.843575199059173)), -x, t_0), (z * -0.0424927283095952));
	} else {
		tmp = x * (4.16438922228 + ((-110.1139242984811 + (3655.1204654076414 / x)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(-0.0424927283095952, y, Float64(z * 0.3041881842569256))
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = Float64(Float64(x * Float64(17.342137594641823 - Float64(12125.076324411626 / Float64(x * x)))) / Float64(4.16438922228 + Float64(110.1139242984811 / x)));
	elseif (x <= 0.4)
		tmp = fma(x, fma(fma(z, -0.38999068429136097, fma(t_0, 7.158593866711955, 5.843575199059173)), Float64(-x), t_0), Float64(z * -0.0424927283095952));
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(Float64(-110.1139242984811 + Float64(3655.1204654076414 / x)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.0424927283095952 * y + N[(z * 0.3041881842569256), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -43000000000.0], N[(N[(x * N[(17.342137594641823 - N[(12125.076324411626 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.16438922228 + N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.4], N[(x * N[(N[(z * -0.38999068429136097 + N[(t$95$0 * 7.158593866711955 + 5.843575199059173), $MachinePrecision]), $MachinePrecision] * (-x) + t$95$0), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * N[(4.16438922228 + N[(N[(-110.1139242984811 + N[(3655.1204654076414 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e10

   1. Initial program 13.4%

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

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \left(\color{blue}{x \cdot \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}, \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

   7. Applied rewrites 22.9%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 90.0

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

   10. Applied rewrites 90.0%

       \[\leadsto \color{blue}{x \cdot \left(4.16438922228 - \frac{110.1139242984811}{x}\right)} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. *-commutative N/A

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

       4. lift--.f64 N/A

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

       5. flip-- N/A

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

       6. associate-*l/ N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. metadata-eval N/A

           \[\leadsto \frac{\left(\color{blue}{\frac{10838835996651139530249}{625000000000000000000}} - \frac{\frac{13764240537310136880149}{125000000000000000000}}{x} \cdot \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \cdot \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       12. lift-/.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \frac{\frac{13764240537310136880149}{125000000000000000000}}{x} \cdot \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000}}{x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       13. frac-times N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{13764240537310136880149}{125000000000000000000}}{x \cdot x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       14. lift-*.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \frac{\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{13764240537310136880149}{125000000000000000000}}{\color{blue}{x \cdot x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       15. lower-/.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{13764240537310136880149}{125000000000000000000}}{x \cdot x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       16. metadata-eval N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \frac{\color{blue}{\frac{189454317568931645604627337684454555190262201}{15625000000000000000000000000000000000000}}}{x \cdot x}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       17. lower-+.f64 90.3

           \[\leadsto \frac{\left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right) \cdot x}{\color{blue}{4.16438922228 + \frac{110.1139242984811}{x}}} \]

   12. Applied rewrites 90.3%

       \[\leadsto \color{blue}{\frac{\left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right) \cdot x}{4.16438922228 + \frac{110.1139242984811}{x}}} \]

   if -4.3e10 < x < 0.40000000000000002

   1. Initial program 99.6%

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\left(-1 \cdot \left(x \cdot \left(\frac{137519416416}{23533438303} + \left(\frac{-215985700909750000000}{553822718361107519809} \cdot z + \frac{336932654197}{47066876606} \cdot \left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right)\right)\right)\right) + \frac{-1000000000}{23533438303} \cdot y\right) - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\left(-1 \cdot \left(x \cdot \left(\frac{137519416416}{23533438303} + \left(\frac{-215985700909750000000}{553822718361107519809} \cdot z + \frac{336932654197}{47066876606} \cdot \left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right)\right)\right)\right) + \frac{-1000000000}{23533438303} \cdot y\right) - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right) + \frac{-1000000000}{23533438303} \cdot z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(-1 \cdot \left(x \cdot \left(\frac{137519416416}{23533438303} + \left(\frac{-215985700909750000000}{553822718361107519809} \cdot z + \frac{336932654197}{47066876606} \cdot \left(\frac{-1000000000}{23533438303} \cdot y - \frac{-168466327098500000000}{553822718361107519809} \cdot z\right)\right)\right)\right) + \frac{-1000000000}{23533438303} \cdot y\right) - \frac{-168466327098500000000}{553822718361107519809} \cdot z, \frac{-1000000000}{23533438303} \cdot z\right)} \]

   7. Applied rewrites 96.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(z, -0.38999068429136097, \mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, z \cdot 0.3041881842569256\right), 7.158593866711955, 5.843575199059173\right)\right), -x, \mathsf{fma}\left(-0.0424927283095952, y, z \cdot 0.3041881842569256\right)\right), -0.0424927283095952 \cdot z\right)} \]

   if 0.40000000000000002 < x

   1. Initial program 17.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-+.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 88.1

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

   5. Applied rewrites 88.1%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\frac{x \cdot \left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right)}{4.16438922228 + \frac{110.1139242984811}{x}}\\ \mathbf{elif}\;x \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(z, -0.38999068429136097, \mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, z \cdot 0.3041881842569256\right), 7.158593866711955, 5.843575199059173\right)\right), -x, \mathsf{fma}\left(-0.0424927283095952, y, z \cdot 0.3041881842569256\right)\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -32000000000000:\\ \;\;\;\;\frac{x \cdot \left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right)}{4.16438922228 + \frac{110.1139242984811}{x}}\\ \mathbf{elif}\;x \leq 112000000:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, y, z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -32000000000000.0)
   (/
    (* x (- 17.342137594641823 (/ 12125.076324411626 (* x x))))
    (+ 4.16438922228 (/ 110.1139242984811 x)))
   (if (<= x 112000000.0)
     (/
      (* (- x 2.0) (fma x y z))
      (+
       (*
        x
        (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
       47.066876606))
     (*
      x
      (+
       4.16438922228
       (/ (+ -110.1139242984811 (/ 3655.1204654076414 x)) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -32000000000000.0) {
		tmp = (x * (17.342137594641823 - (12125.076324411626 / (x * x)))) / (4.16438922228 + (110.1139242984811 / x));
	} else if (x <= 112000000.0) {
		tmp = ((x - 2.0) * fma(x, y, z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606);
	} else {
		tmp = x * (4.16438922228 + ((-110.1139242984811 + (3655.1204654076414 / x)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -32000000000000.0)
		tmp = Float64(Float64(x * Float64(17.342137594641823 - Float64(12125.076324411626 / Float64(x * x)))) / Float64(4.16438922228 + Float64(110.1139242984811 / x)));
	elseif (x <= 112000000.0)
		tmp = Float64(Float64(Float64(x - 2.0) * fma(x, y, z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606));
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(Float64(-110.1139242984811 + Float64(3655.1204654076414 / x)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -32000000000000.0], N[(N[(x * N[(17.342137594641823 - N[(12125.076324411626 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.16438922228 + N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 112000000.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(x * y + 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 * N[(4.16438922228 + N[(N[(-110.1139242984811 + N[(3655.1204654076414 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2e13

   1. Initial program 13.4%

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

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \left(\color{blue}{x \cdot \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}, \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

   7. Applied rewrites 22.9%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 90.0

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

   10. Applied rewrites 90.0%

       \[\leadsto \color{blue}{x \cdot \left(4.16438922228 - \frac{110.1139242984811}{x}\right)} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. *-commutative N/A

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

       4. lift--.f64 N/A

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

       5. flip-- N/A

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

       6. associate-*l/ N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. metadata-eval N/A

           \[\leadsto \frac{\left(\color{blue}{\frac{10838835996651139530249}{625000000000000000000}} - \frac{\frac{13764240537310136880149}{125000000000000000000}}{x} \cdot \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \cdot \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       12. lift-/.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \frac{\frac{13764240537310136880149}{125000000000000000000}}{x} \cdot \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000}}{x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       13. frac-times N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{13764240537310136880149}{125000000000000000000}}{x \cdot x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       14. lift-*.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \frac{\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{13764240537310136880149}{125000000000000000000}}{\color{blue}{x \cdot x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       15. lower-/.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{13764240537310136880149}{125000000000000000000}}{x \cdot x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       16. metadata-eval N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \frac{\color{blue}{\frac{189454317568931645604627337684454555190262201}{15625000000000000000000000000000000000000}}}{x \cdot x}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       17. lower-+.f64 90.3

           \[\leadsto \frac{\left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right) \cdot x}{\color{blue}{4.16438922228 + \frac{110.1139242984811}{x}}} \]

   12. Applied rewrites 90.3%

       \[\leadsto \color{blue}{\frac{\left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right) \cdot x}{4.16438922228 + \frac{110.1139242984811}{x}}} \]

   if -3.2e13 < x < 1.12e8

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 93.6

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

   5. Applied rewrites 93.6%

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

   if 1.12e8 < x

   1. Initial program 15.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-+.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -32000000000000:\\ \;\;\;\;\frac{x \cdot \left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right)}{4.16438922228 + \frac{110.1139242984811}{x}}\\ \mathbf{elif}\;x \leq 112000000:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, y, z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\frac{x \cdot \left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right)}{4.16438922228 + \frac{110.1139242984811}{x}}\\ \mathbf{elif}\;x \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(x, 9.590778533639166 \cdot 10^{-6} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(z, -186384.13759495187, \mathsf{fma}\left(y, 31716.735319244533, -609291.0162155379\right)\right), \mathsf{fma}\left(y, -4430.58174688886, z \cdot 31716.735319244533\right)\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -43000000000.0)
   (/
    (* x (- 17.342137594641823 (/ 12125.076324411626 (* x x))))
    (+ 4.16438922228 (/ 110.1139242984811 x)))
   (if (<= x 0.4)
     (fma
      x
      (*
       9.590778533639166e-6
       (fma
        x
        (fma
         z
         -186384.13759495187
         (fma y 31716.735319244533 -609291.0162155379))
        (fma y -4430.58174688886 (* z 31716.735319244533))))
      (* z -0.0424927283095952))
     (*
      x
      (+
       4.16438922228
       (/ (+ -110.1139242984811 (/ 3655.1204654076414 x)) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -43000000000.0) {
		tmp = (x * (17.342137594641823 - (12125.076324411626 / (x * x)))) / (4.16438922228 + (110.1139242984811 / x));
	} else if (x <= 0.4) {
		tmp = fma(x, (9.590778533639166e-6 * fma(x, fma(z, -186384.13759495187, fma(y, 31716.735319244533, -609291.0162155379)), fma(y, -4430.58174688886, (z * 31716.735319244533)))), (z * -0.0424927283095952));
	} else {
		tmp = x * (4.16438922228 + ((-110.1139242984811 + (3655.1204654076414 / x)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = Float64(Float64(x * Float64(17.342137594641823 - Float64(12125.076324411626 / Float64(x * x)))) / Float64(4.16438922228 + Float64(110.1139242984811 / x)));
	elseif (x <= 0.4)
		tmp = fma(x, Float64(9.590778533639166e-6 * fma(x, fma(z, -186384.13759495187, fma(y, 31716.735319244533, -609291.0162155379)), fma(y, -4430.58174688886, Float64(z * 31716.735319244533)))), Float64(z * -0.0424927283095952));
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(Float64(-110.1139242984811 + Float64(3655.1204654076414 / x)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -43000000000.0], N[(N[(x * N[(17.342137594641823 - N[(12125.076324411626 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.16438922228 + N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.4], N[(x * N[(9.590778533639166e-6 * N[(x * N[(z * -186384.13759495187 + N[(y * 31716.735319244533 + -609291.0162155379), $MachinePrecision]), $MachinePrecision] + N[(y * -4430.58174688886 + N[(z * 31716.735319244533), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * N[(4.16438922228 + N[(N[(-110.1139242984811 + N[(3655.1204654076414 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e10

   1. Initial program 13.4%

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

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \left(\color{blue}{x \cdot \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}, \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

   7. Applied rewrites 22.9%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 90.0

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

   10. Applied rewrites 90.0%

       \[\leadsto \color{blue}{x \cdot \left(4.16438922228 - \frac{110.1139242984811}{x}\right)} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. *-commutative N/A

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

       4. lift--.f64 N/A

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

       5. flip-- N/A

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

       6. associate-*l/ N/A

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

       7. lower-/.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lower--.f64 N/A

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

       10. metadata-eval N/A

           \[\leadsto \frac{\left(\color{blue}{\frac{10838835996651139530249}{625000000000000000000}} - \frac{\frac{13764240537310136880149}{125000000000000000000}}{x} \cdot \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       11. lift-/.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \cdot \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       12. lift-/.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \frac{\frac{13764240537310136880149}{125000000000000000000}}{x} \cdot \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000}}{x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       13. frac-times N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{13764240537310136880149}{125000000000000000000}}{x \cdot x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       14. lift-*.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \frac{\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{13764240537310136880149}{125000000000000000000}}{\color{blue}{x \cdot x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       15. lower-/.f64 N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \color{blue}{\frac{\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{13764240537310136880149}{125000000000000000000}}{x \cdot x}}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       16. metadata-eval N/A

           \[\leadsto \frac{\left(\frac{10838835996651139530249}{625000000000000000000} - \frac{\color{blue}{\frac{189454317568931645604627337684454555190262201}{15625000000000000000000000000000000000000}}}{x \cdot x}\right) \cdot x}{\frac{104109730557}{25000000000} + \frac{\frac{13764240537310136880149}{125000000000000000000}}{x}} \]

       17. lower-+.f64 90.3

           \[\leadsto \frac{\left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right) \cdot x}{\color{blue}{4.16438922228 + \frac{110.1139242984811}{x}}} \]

   12. Applied rewrites 90.3%

       \[\leadsto \color{blue}{\frac{\left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right) \cdot x}{4.16438922228 + \frac{110.1139242984811}{x}}} \]

   if -4.3e10 < x < 0.40000000000000002

   1. Initial program 99.6%

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

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right)\right)\right), 104266.82218680695\right)} \cdot \frac{x + -2}{\frac{1}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 2215.29087344443\right) - x \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right) \cdot 47.066876606\right)}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\frac{125000000000000000000000000}{13033352773350869092174451844127} \cdot \left(x \cdot \left(\left(\frac{-5824504299842245708137}{31250000000000000} \cdot z + \frac{7929183829811133507691}{250000000000000000} \cdot y\right) - \frac{2380043032091944814861338743267}{3906250000000000000000000}\right)\right) + \frac{125000000000000000000000000}{13033352773350869092174451844127} \cdot \left(\frac{-553822718361107519809}{125000000000000000} \cdot y + \frac{7929183829811133507691}{250000000000000000} \cdot z\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{125000000000000000000000000}{13033352773350869092174451844127} \cdot \left(x \cdot \left(\left(\frac{-5824504299842245708137}{31250000000000000} \cdot z + \frac{7929183829811133507691}{250000000000000000} \cdot y\right) - \frac{2380043032091944814861338743267}{3906250000000000000000000}\right)\right) + \frac{125000000000000000000000000}{13033352773350869092174451844127} \cdot \left(\frac{-553822718361107519809}{125000000000000000} \cdot y + \frac{7929183829811133507691}{250000000000000000} \cdot z\right)\right) + \frac{-1000000000}{23533438303} \cdot z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{125000000000000000000000000}{13033352773350869092174451844127} \cdot \left(x \cdot \left(\left(\frac{-5824504299842245708137}{31250000000000000} \cdot z + \frac{7929183829811133507691}{250000000000000000} \cdot y\right) - \frac{2380043032091944814861338743267}{3906250000000000000000000}\right)\right) + \frac{125000000000000000000000000}{13033352773350869092174451844127} \cdot \left(\frac{-553822718361107519809}{125000000000000000} \cdot y + \frac{7929183829811133507691}{250000000000000000} \cdot z\right), \frac{-1000000000}{23533438303} \cdot z\right)} \]

   7. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 9.590778533639166 \cdot 10^{-6} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(z, -186384.13759495187, \mathsf{fma}\left(y, 31716.735319244533, -609291.0162155379\right)\right), \mathsf{fma}\left(y, -4430.58174688886, z \cdot 31716.735319244533\right)\right), -0.0424927283095952 \cdot z\right)} \]

   if 0.40000000000000002 < x

   1. Initial program 17.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-+.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 88.1

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

   5. Applied rewrites 88.1%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\frac{x \cdot \left(17.342137594641823 - \frac{12125.076324411626}{x \cdot x}\right)}{4.16438922228 + \frac{110.1139242984811}{x}}\\ \mathbf{elif}\;x \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(x, 9.590778533639166 \cdot 10^{-6} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(z, -186384.13759495187, \mathsf{fma}\left(y, 31716.735319244533, -609291.0162155379\right)\right), \mathsf{fma}\left(y, -4430.58174688886, z \cdot 31716.735319244533\right)\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(x, 9.590778533639166 \cdot 10^{-6} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(z, -186384.13759495187, \mathsf{fma}\left(y, 31716.735319244533, -609291.0162155379\right)\right), \mathsf{fma}\left(y, -4430.58174688886, z \cdot 31716.735319244533\right)\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -43000000000.0)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 0.4)
     (fma
      x
      (*
       9.590778533639166e-6
       (fma
        x
        (fma
         z
         -186384.13759495187
         (fma y 31716.735319244533 -609291.0162155379))
        (fma y -4430.58174688886 (* z 31716.735319244533))))
      (* z -0.0424927283095952))
     (*
      x
      (+
       4.16438922228
       (/ (+ -110.1139242984811 (/ 3655.1204654076414 x)) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -43000000000.0) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 0.4) {
		tmp = fma(x, (9.590778533639166e-6 * fma(x, fma(z, -186384.13759495187, fma(y, 31716.735319244533, -609291.0162155379)), fma(y, -4430.58174688886, (z * 31716.735319244533)))), (z * -0.0424927283095952));
	} else {
		tmp = x * (4.16438922228 + ((-110.1139242984811 + (3655.1204654076414 / x)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 0.4)
		tmp = fma(x, Float64(9.590778533639166e-6 * fma(x, fma(z, -186384.13759495187, fma(y, 31716.735319244533, -609291.0162155379)), fma(y, -4430.58174688886, Float64(z * 31716.735319244533)))), Float64(z * -0.0424927283095952));
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(Float64(-110.1139242984811 + Float64(3655.1204654076414 / x)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -43000000000.0], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 0.4], N[(x * N[(9.590778533639166e-6 * N[(x * N[(z * -186384.13759495187 + N[(y * 31716.735319244533 + -609291.0162155379), $MachinePrecision]), $MachinePrecision] + N[(y * -4430.58174688886 + N[(z * 31716.735319244533), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * N[(4.16438922228 + N[(N[(-110.1139242984811 + N[(3655.1204654076414 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e10

   1. Initial program 13.4%

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

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \left(\color{blue}{x \cdot \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}, \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

   7. Applied rewrites 22.9%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 90.0

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

   10. Applied rewrites 90.0%

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

   11. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 90.0

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

   13. Applied rewrites 90.0%

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

   if -4.3e10 < x < 0.40000000000000002

   1. Initial program 99.6%

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

   4. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right) \cdot \left(x \cdot \left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right)\right)\right), 104266.82218680695\right)} \cdot \frac{x + -2}{\frac{1}{\mathsf{fma}\left(x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 2215.29087344443\right) - x \cdot \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right) \cdot 47.066876606\right)}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\frac{125000000000000000000000000}{13033352773350869092174451844127} \cdot \left(x \cdot \left(\left(\frac{-5824504299842245708137}{31250000000000000} \cdot z + \frac{7929183829811133507691}{250000000000000000} \cdot y\right) - \frac{2380043032091944814861338743267}{3906250000000000000000000}\right)\right) + \frac{125000000000000000000000000}{13033352773350869092174451844127} \cdot \left(\frac{-553822718361107519809}{125000000000000000} \cdot y + \frac{7929183829811133507691}{250000000000000000} \cdot z\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{125000000000000000000000000}{13033352773350869092174451844127} \cdot \left(x \cdot \left(\left(\frac{-5824504299842245708137}{31250000000000000} \cdot z + \frac{7929183829811133507691}{250000000000000000} \cdot y\right) - \frac{2380043032091944814861338743267}{3906250000000000000000000}\right)\right) + \frac{125000000000000000000000000}{13033352773350869092174451844127} \cdot \left(\frac{-553822718361107519809}{125000000000000000} \cdot y + \frac{7929183829811133507691}{250000000000000000} \cdot z\right)\right) + \frac{-1000000000}{23533438303} \cdot z} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{125000000000000000000000000}{13033352773350869092174451844127} \cdot \left(x \cdot \left(\left(\frac{-5824504299842245708137}{31250000000000000} \cdot z + \frac{7929183829811133507691}{250000000000000000} \cdot y\right) - \frac{2380043032091944814861338743267}{3906250000000000000000000}\right)\right) + \frac{125000000000000000000000000}{13033352773350869092174451844127} \cdot \left(\frac{-553822718361107519809}{125000000000000000} \cdot y + \frac{7929183829811133507691}{250000000000000000} \cdot z\right), \frac{-1000000000}{23533438303} \cdot z\right)} \]

   7. Applied rewrites 93.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 9.590778533639166 \cdot 10^{-6} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(z, -186384.13759495187, \mathsf{fma}\left(y, 31716.735319244533, -609291.0162155379\right)\right), \mathsf{fma}\left(y, -4430.58174688886, z \cdot 31716.735319244533\right)\right), -0.0424927283095952 \cdot z\right)} \]

   if 0.40000000000000002 < x

   1. Initial program 17.1%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-+.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 88.1

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

   5. Applied rewrites 88.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 0.4:\\ \;\;\;\;\mathsf{fma}\left(x, 9.590778533639166 \cdot 10^{-6} \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(z, -186384.13759495187, \mathsf{fma}\left(y, 31716.735319244533, -609291.0162155379\right)\right), \mathsf{fma}\left(y, -4430.58174688886, z \cdot 31716.735319244533\right)\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;z \cdot \left(-0.0424927283095952 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), \frac{x}{z}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -43000000000.0)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 2.0)
     (*
      z
      (*
       -0.0424927283095952
       (fma
        (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
        (/ x z)
        1.0)))
     (*
      x
      (+
       4.16438922228
       (/ (+ -110.1139242984811 (/ 3655.1204654076414 x)) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -43000000000.0) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 2.0) {
		tmp = z * (-0.0424927283095952 * fma(fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), (x / z), 1.0));
	} else {
		tmp = x * (4.16438922228 + ((-110.1139242984811 + (3655.1204654076414 / x)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 2.0)
		tmp = Float64(z * Float64(-0.0424927283095952 * fma(fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), Float64(x / z), 1.0)));
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(Float64(-110.1139242984811 + Float64(3655.1204654076414 / x)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -43000000000.0], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 2.0], N[(z * N[(-0.0424927283095952 * N[(N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] * N[(x / z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(4.16438922228 + N[(N[(-110.1139242984811 + N[(3655.1204654076414 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e10

   1. Initial program 13.4%

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

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \left(\color{blue}{x \cdot \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}, \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

   7. Applied rewrites 22.9%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 90.0

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

   10. Applied rewrites 90.0%

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

   11. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 90.0

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

   13. Applied rewrites 90.0%

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

   if -4.3e10 < x < 2

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 94.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), \frac{x}{z}, 1\right) \cdot \color{blue}{\frac{-1000000000}{23533438303}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 91.8%

      \[\leadsto z \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), \frac{x}{z}, 1\right) \cdot \color{blue}{-0.0424927283095952}\right) \]

   if 2 < x

   1. Initial program 15.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-+.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;z \cdot \left(-0.0424927283095952 \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), \frac{x}{z}, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 4.4:\\ \;\;\;\;\mathsf{fma}\left(z, -0.0424927283095952, x \cdot \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), z \cdot 0.28294182010212804\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -43000000000.0)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 4.4)
     (fma
      z
      -0.0424927283095952
      (* x (fma 0.0212463641547976 (fma y -2.0 z) (* z 0.28294182010212804))))
     (*
      x
      (+
       4.16438922228
       (/ (+ -110.1139242984811 (/ 3655.1204654076414 x)) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -43000000000.0) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 4.4) {
		tmp = fma(z, -0.0424927283095952, (x * fma(0.0212463641547976, fma(y, -2.0, z), (z * 0.28294182010212804))));
	} else {
		tmp = x * (4.16438922228 + ((-110.1139242984811 + (3655.1204654076414 / x)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 4.4)
		tmp = fma(z, -0.0424927283095952, Float64(x * fma(0.0212463641547976, fma(y, -2.0, z), Float64(z * 0.28294182010212804))));
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(Float64(-110.1139242984811 + Float64(3655.1204654076414 / x)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -43000000000.0], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 4.4], N[(z * -0.0424927283095952 + N[(x * N[(0.0212463641547976 * N[(y * -2.0 + z), $MachinePrecision] + N[(z * 0.28294182010212804), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(4.16438922228 + N[(N[(-110.1139242984811 + N[(3655.1204654076414 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e10

   1. Initial program 13.4%

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

   4. Applied rewrites 19.6%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \left(\color{blue}{x \cdot \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}, \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

   7. Applied rewrites 22.9%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 90.0

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

   10. Applied rewrites 90.0%

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

   11. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 90.0

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

   13. Applied rewrites 90.0%

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

   if -4.3e10 < x < 4.4000000000000004

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. flip-+ N/A

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

      10. div-sub N/A

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 89.9

          \[\leadsto \mathsf{fma}\left(z, -0.0424927283095952, x \cdot \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), \color{blue}{0.28294182010212804} \cdot z\right)\right) \]

   7. Applied rewrites 89.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, -0.0424927283095952, x \cdot \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), 0.28294182010212804 \cdot z\right)\right)} \]

   if 4.4000000000000004 < x

   1. Initial program 15.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-+.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 4.4:\\ \;\;\;\;\mathsf{fma}\left(z, -0.0424927283095952, x \cdot \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), z \cdot 0.28294182010212804\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 27:\\ \;\;\;\;\mathsf{fma}\left(z, -0.0424927283095952, x \cdot \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), z \cdot 0.28294182010212804\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -43000000000.0)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 27.0)
     (fma
      z
      -0.0424927283095952
      (* x (fma 0.0212463641547976 (fma y -2.0 z) (* z 0.28294182010212804))))
     (fma x 4.16438922228 -110.1139242984811))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -43000000000.0) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 27.0) {
		tmp = fma(z, -0.0424927283095952, (x * fma(0.0212463641547976, fma(y, -2.0, z), (z * 0.28294182010212804))));
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 27.0)
		tmp = fma(z, -0.0424927283095952, Float64(x * fma(0.0212463641547976, fma(y, -2.0, z), Float64(z * 0.28294182010212804))));
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -43000000000.0], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 27.0], N[(z * -0.0424927283095952 + N[(x * N[(0.0212463641547976 * N[(y * -2.0 + z), $MachinePrecision] + N[(z * 0.28294182010212804), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3e10 or 27 < x

   1. Initial program 14.8%

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

   4. Applied rewrites 19.1%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \left(\color{blue}{x \cdot \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}, \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

   7. Applied rewrites 21.5%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 89.5

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

   10. Applied rewrites 89.5%

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

   11. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 89.5

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

   13. Applied rewrites 89.5%

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

   if -4.3e10 < x < 27

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. flip-+ N/A

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

      10. div-sub N/A

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

   4. Applied rewrites 63.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval 89.9

          \[\leadsto \mathsf{fma}\left(z, -0.0424927283095952, x \cdot \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), \color{blue}{0.28294182010212804} \cdot z\right)\right) \]

   7. Applied rewrites 89.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 27:\\ \;\;\;\;\mathsf{fma}\left(z, -0.0424927283095952, x \cdot \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), z \cdot 0.28294182010212804\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 27:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, z \cdot 0.3041881842569256\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -43000000000.0)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 27.0)
     (fma
      x
      (fma -0.0424927283095952 y (* z 0.3041881842569256))
      (* z -0.0424927283095952))
     (fma x 4.16438922228 -110.1139242984811))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -43000000000.0) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 27.0) {
		tmp = fma(x, fma(-0.0424927283095952, y, (z * 0.3041881842569256)), (z * -0.0424927283095952));
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 27.0)
		tmp = fma(x, fma(-0.0424927283095952, y, Float64(z * 0.3041881842569256)), Float64(z * -0.0424927283095952));
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -43000000000.0], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 27.0], N[(x * N[(-0.0424927283095952 * y + N[(z * 0.3041881842569256), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3e10 or 27 < x

   1. Initial program 14.8%

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

   4. Applied rewrites 19.1%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \left(\color{blue}{x \cdot \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}, \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

   7. Applied rewrites 21.5%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 89.5

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

   10. Applied rewrites 89.5%

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

   11. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 89.5

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

   13. Applied rewrites 89.5%

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

   if -4.3e10 < x < 27

   1. Initial program 99.6%

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   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. lower-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 89.9

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

   7. Applied rewrites 89.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 27:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, z \cdot 0.3041881842569256\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-23}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.029)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 5.8e-23)
     (* z -0.0424927283095952)
     (fma x 4.16438922228 -110.1139242984811))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.029) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 5.8e-23) {
		tmp = z * -0.0424927283095952;
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.029)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 5.8e-23)
		tmp = Float64(z * -0.0424927283095952);
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.029], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 5.8e-23], N[(z * -0.0424927283095952), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0290000000000000015 or 5.8000000000000003e-23 < x

   1. Initial program 19.1%

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

   4. Applied rewrites 23.2%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \left(\color{blue}{x \cdot \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}, \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

   7. Applied rewrites 25.5%

      \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 85.2

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

   10. Applied rewrites 85.2%

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

   11. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 85.3

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

   13. Applied rewrites 85.3%

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

   if -0.0290000000000000015 < x < 5.8000000000000003e-23

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot \frac{-1000000000}{23533438303}} \]

      2. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

Alternative 15: 76.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 0.4:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.029)
   (* x 4.16438922228)
   (if (<= x 0.4) (* z -0.0424927283095952) (* x 4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.029) {
		tmp = x * 4.16438922228;
	} else if (x <= 0.4) {
		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 <= (-0.029d0)) then
        tmp = x * 4.16438922228d0
    else if (x <= 0.4d0) 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 <= -0.029) {
		tmp = x * 4.16438922228;
	} else if (x <= 0.4) {
		tmp = z * -0.0424927283095952;
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.029:
		tmp = x * 4.16438922228
	elif x <= 0.4:
		tmp = z * -0.0424927283095952
	else:
		tmp = x * 4.16438922228
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.029)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= 0.4)
		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 <= -0.029)
		tmp = x * 4.16438922228;
	elseif (x <= 0.4)
		tmp = z * -0.0424927283095952;
	else
		tmp = x * 4.16438922228;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.029], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, 0.4], N[(z * -0.0424927283095952), $MachinePrecision], N[(x * 4.16438922228), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0290000000000000015 or 0.40000000000000002 < x

   1. Initial program 17.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   if -0.0290000000000000015 < x < 0.40000000000000002

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot \frac{-1000000000}{23533438303}} \]

      2. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

Alternative 16: 45.7% 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 56.5%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 46.6

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

5. Applied rewrites 46.6%

   \[\leadsto \color{blue}{x \cdot 4.16438922228} \]
6. Add Preprocessing

Alternative 17: 3.3% accurate, 79.0× speedup

Math

\[\begin{array}{l} \\ -110.1139242984811 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -110.1139242984811)

C

double code(double x, double y, double z) {
	return -110.1139242984811;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -110.1139242984811d0
end function

Java

public static double code(double x, double y, double z) {
	return -110.1139242984811;
}

Python

def code(x, y, z):
	return -110.1139242984811

Julia

function code(x, y, z)
	return -110.1139242984811
end

MATLAB

function tmp = code(x, y, z)
	tmp = -110.1139242984811;
end

Wolfram

code[x_, y_, z_] := -110.1139242984811

Derivation

1. Initial program 56.5%

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

4. Applied rewrites 58.7%

   \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)}} \]

5. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\left(\frac{x \cdot \left(y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

   2. associate-/l* N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \left(\color{blue}{x \cdot \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}} + \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{y + x \cdot \left(\frac{4297481763}{31250000} + x \cdot \left(\frac{393497462077}{5000000000} + \frac{104109730557}{25000000000} \cdot x\right)\right)}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}, \frac{z}{\frac{23533438303}{500000000} + x \cdot \left(\frac{156699607947}{500000000} + x \cdot \left(\frac{263505074721}{1000000000} + x \cdot \left(\frac{216700011257}{5000000000} + x\right)\right)\right)}\right)}\right)}{\mathsf{neg}\left(\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)\right)} \]

7. Applied rewrites 59.9%

   \[\leadsto \frac{-\mathsf{fma}\left(x, x \cdot x, -8\right) \cdot \color{blue}{\mathsf{fma}\left(x, \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}, \frac{z}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)}}{-\mathsf{fma}\left(x, x, \mathsf{fma}\left(x, 2, 4\right)\right)} \]

8. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f64 47.3

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

10. Applied rewrites 47.3%

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

11. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{-13764240537310136880149}{125000000000000000000}} \]
12. Step-by-step derivation

13. Applied rewrites 3.4%

    \[\leadsto \color{blue}{-110.1139242984811} \]
14. 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 2024220 
(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)))