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

Percentage Accurate: 59.6% → 98.6%

Time: 17.0s

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: 59.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \frac{\frac{3655.1204654076414}{x} + -110.1139242984811}{x}\\ t_2 := \frac{130977.50649958357 - y}{t\_0} - t\_1\\ \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq \infty:\\ \;\;\;\;\left(\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right) \cdot \frac{1}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(17.342137594641823 + \left(t\_1 + \frac{y - 130977.50649958357}{t\_0}\right) \cdot t\_2\right)}{4.16438922228 + t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (/ (+ (/ 3655.1204654076414 x) -110.1139242984811) x))
        (t_2 (- (/ (- 130977.50649958357 y) t_0) t_1)))
   (if (<=
        (/
         (*
          (- x 2.0)
          (+
           (*
            x
            (+
             (*
              x
              (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
             y))
           z))
         (+
          (*
           x
           (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
          47.066876606))
        INFINITY)
     (*
      (*
       (fma x x -4.0)
       (/
        (fma
         x
         (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
         z)
        (fma
         x
         (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
         47.066876606)))
      (/ 1.0 (+ x 2.0)))
     (/
      (*
       x
       (+ 17.342137594641823 (* (+ t_1 (/ (- y 130977.50649958357) t_0)) t_2)))
      (+ 4.16438922228 t_2)))))

C

double code(double x, double y, double z) {
	double t_0 = x * (x * x);
	double t_1 = ((3655.1204654076414 / x) + -110.1139242984811) / x;
	double t_2 = ((130977.50649958357 - y) / t_0) - t_1;
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= ((double) INFINITY)) {
		tmp = (fma(x, x, -4.0) * (fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606))) * (1.0 / (x + 2.0));
	} else {
		tmp = (x * (17.342137594641823 + ((t_1 + ((y - 130977.50649958357) / t_0)) * t_2))) / (4.16438922228 + t_2);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(x * x))
	t_1 = Float64(Float64(Float64(3655.1204654076414 / x) + -110.1139242984811) / x)
	t_2 = Float64(Float64(Float64(130977.50649958357 - y) / t_0) - t_1)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= Inf)
		tmp = Float64(Float64(fma(x, x, -4.0) * Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606))) * Float64(1.0 / Float64(x + 2.0)));
	else
		tmp = Float64(Float64(x * Float64(17.342137594641823 + Float64(Float64(t_1 + Float64(Float64(y - 130977.50649958357) / t_0)) * t_2))) / Float64(4.16438922228 + t_2));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] + -110.1139242984811), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(130977.50649958357 - y), $MachinePrecision] / t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x * x + -4.0), $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(17.342137594641823 + N[(N[(t$95$1 + N[(N[(y - 130977.50649958357), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.16438922228 + t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64))) < +inf.0

   1. Initial program 94.1%

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

      1. associate-/l* N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   if +inf.0 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} + \frac{y}{{x}^{3}}\right)\right) - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{{x}^{3}}\right)\right)} \]

   5. Simplified 99.1%

      \[\leadsto \color{blue}{x \cdot \left(4.16438922228 + \left(\frac{y}{x \cdot \left(x \cdot x\right)} + \left(\frac{\frac{3655.1204654076414}{x} + -110.1139242984811}{x} - \frac{130977.50649958357}{x \cdot \left(x \cdot x\right)}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. flip-+ N/A

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

      3. associate-*l/ N/A

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

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

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

   7. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\left(17.342137594641823 - \left(\frac{\frac{3655.1204654076414}{x} + -110.1139242984811}{x} - \frac{130977.50649958357 - y}{x \cdot \left(x \cdot x\right)}\right) \cdot \left(\frac{\frac{3655.1204654076414}{x} + -110.1139242984811}{x} - \frac{130977.50649958357 - y}{x \cdot \left(x \cdot x\right)}\right)\right) \cdot x}{4.16438922228 - \left(\frac{\frac{3655.1204654076414}{x} + -110.1139242984811}{x} - \frac{130977.50649958357 - y}{x \cdot \left(x \cdot x\right)}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq \infty:\\ \;\;\;\;\left(\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right) \cdot \frac{1}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(17.342137594641823 + \left(\frac{\frac{3655.1204654076414}{x} + -110.1139242984811}{x} + \frac{y - 130977.50649958357}{x \cdot \left(x \cdot x\right)}\right) \cdot \left(\frac{130977.50649958357 - y}{x \cdot \left(x \cdot x\right)} - \frac{\frac{3655.1204654076414}{x} + -110.1139242984811}{x}\right)\right)}{4.16438922228 + \left(\frac{130977.50649958357 - y}{x \cdot \left(x \cdot x\right)} - \frac{\frac{3655.1204654076414}{x} + -110.1139242984811}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.5× speedup

Math

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

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))
      INFINITY)
   (*
    (*
     (fma x x -4.0)
     (/
      (fma
       x
       (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
       z)
      (fma
       x
       (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
       47.066876606)))
    (/ 1.0 (+ x 2.0)))
   (* x (+ 4.16438922228 (/ y (* x (* x x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= ((double) INFINITY)) {
		tmp = (fma(x, x, -4.0) * (fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606))) * (1.0 / (x + 2.0));
	} else {
		tmp = x * (4.16438922228 + (y / (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= Inf)
		tmp = Float64(Float64(fma(x, x, -4.0) * Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606))) * Float64(1.0 / Float64(x + 2.0)));
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(y / Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(x * x + -4.0), $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(4.16438922228 + N[(y / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64))) < +inf.0

   1. Initial program 94.1%

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

      1. associate-/l* N/A

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

      2. flip-- N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   if +inf.0 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} + \frac{y}{{x}^{3}}\right)\right) - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{{x}^{3}}\right)\right)} \]

   5. Simplified 99.1%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 99.1

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

   8. Simplified 99.1%

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

4. Final simplification 99.3%

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

Alternative 3: 98.7% accurate, 0.5× speedup

Math

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

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64))) < +inf.0

   1. Initial program 94.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   if +inf.0 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} + \frac{y}{{x}^{3}}\right)\right) - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{{x}^{3}}\right)\right)} \]

   5. Simplified 99.1%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 99.1

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

   8. Simplified 99.1%

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

4. Final simplification 99.3%

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

Alternative 4: 93.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(x, y, z\right) \cdot \frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-110.1139242984811 + \frac{\left(\frac{y}{x} - -3655.1204654076414\right) - \frac{130977.50649958357}{x}}{x}}{x} - -4.16438922228\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4.6e+19)
   (* x (+ 4.16438922228 (/ y (* x (* x x)))))
   (if (<= x 3.1e+40)
     (*
      (fma x y z)
      (/
       (+ x -2.0)
       (fma
        x
        (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
        47.066876606)))
     (*
      x
      (-
       (/
        (+
         -110.1139242984811
         (/ (- (- (/ y x) -3655.1204654076414) (/ 130977.50649958357 x)) x))
        x)
       -4.16438922228)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4.6e+19) {
		tmp = x * (4.16438922228 + (y / (x * (x * x))));
	} else if (x <= 3.1e+40) {
		tmp = fma(x, y, z) * ((x + -2.0) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606));
	} else {
		tmp = x * (((-110.1139242984811 + ((((y / x) - -3655.1204654076414) - (130977.50649958357 / x)) / x)) / x) - -4.16438922228);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4.6e+19)
		tmp = Float64(x * Float64(4.16438922228 + Float64(y / Float64(x * Float64(x * x)))));
	elseif (x <= 3.1e+40)
		tmp = Float64(fma(x, y, z) * Float64(Float64(x + -2.0) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)));
	else
		tmp = Float64(x * Float64(Float64(Float64(-110.1139242984811 + Float64(Float64(Float64(Float64(y / x) - -3655.1204654076414) - Float64(130977.50649958357 / x)) / x)) / x) - -4.16438922228));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4.6e+19], N[(x * N[(4.16438922228 + N[(y / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.1e+40], N[(N[(x * y + z), $MachinePrecision] * N[(N[(x + -2.0), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(-110.1139242984811 + N[(N[(N[(N[(y / x), $MachinePrecision] - -3655.1204654076414), $MachinePrecision] - N[(130977.50649958357 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.6e19

   1. Initial program 20.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} + \frac{y}{{x}^{3}}\right)\right) - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{{x}^{3}}\right)\right)} \]

   5. Simplified 97.2%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 97.2

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

   8. Simplified 97.2%

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

   if -4.6e19 < x < 3.0999999999999998e40

   1. Initial program 98.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 97.0

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

   7. Simplified 97.0%

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

   if 3.0999999999999998e40 < x

   1. Initial program 9.8%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+19}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(x, y, z\right) \cdot \frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-110.1139242984811 + \frac{\left(\frac{y}{x} - -3655.1204654076414\right) - \frac{130977.50649958357}{x}}{x}}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(4.16438922228 + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{if}\;x \leq -1.16 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+40}:\\ \;\;\;\;\mathsf{fma}\left(x, y, z\right) \cdot \frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ 4.16438922228 (/ y (* x (* x x)))))))
   (if (<= x -1.16e+20)
     t_0
     (if (<= x 3.1e+40)
       (*
        (fma x y z)
        (/
         (+ x -2.0)
         (fma
          x
          (fma x (fma 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 * (4.16438922228 + (y / (x * (x * x))));
	double tmp;
	if (x <= -1.16e+20) {
		tmp = t_0;
	} else if (x <= 3.1e+40) {
		tmp = fma(x, y, z) * ((x + -2.0) / fma(x, fma(x, fma(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(4.16438922228 + Float64(y / Float64(x * Float64(x * x)))))
	tmp = 0.0
	if (x <= -1.16e+20)
		tmp = t_0;
	elseif (x <= 3.1e+40)
		tmp = Float64(fma(x, y, z) * Float64(Float64(x + -2.0) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.16438922228 + N[(y / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.16e+20], t$95$0, If[LessEqual[x, 3.1e+40], N[(N[(x * y + z), $MachinePrecision] * N[(N[(x + -2.0), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.16e20 or 3.0999999999999998e40 < x

   1. Initial program 15.7%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} + \frac{y}{{x}^{3}}\right)\right) - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{{x}^{3}}\right)\right)} \]

   5. Simplified 98.0%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 98.0

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

   8. Simplified 98.0%

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

   if -1.16e20 < x < 3.0999999999999998e40

   1. Initial program 98.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 97.0

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

   7. Simplified 97.0%

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

Alternative 6: 95.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(4.16438922228 + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{if}\;x \leq -3800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.5:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ 4.16438922228 (/ y (* x (* x x)))))))
   (if (<= x -3800000000.0)
     t_0
     (if (<= x 3.5)
       (*
        (fma
         x
         (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
         z)
        (fma x 0.3041881842569256 -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.16438922228 + (y / (x * (x * x))));
	double tmp;
	if (x <= -3800000000.0) {
		tmp = t_0;
	} else if (x <= 3.5) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, 0.3041881842569256, -0.0424927283095952);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.16438922228 + Float64(y / Float64(x * Float64(x * x)))))
	tmp = 0.0
	if (x <= -3800000000.0)
		tmp = t_0;
	elseif (x <= 3.5)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, 0.3041881842569256, -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.16438922228 + N[(y / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3800000000.0], t$95$0, If[LessEqual[x, 3.5], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * N[(x * 0.3041881842569256 + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e9 or 3.5 < x

   1. Initial program 21.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} + \frac{y}{{x}^{3}}\right)\right) - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{{x}^{3}}\right)\right)} \]

   5. Simplified 93.8%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 93.3

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

   8. Simplified 93.3%

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

   if -3.8e9 < x < 3.5

   1. Initial program 99.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 96.8

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

   7. Simplified 96.8%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \color{blue}{\mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(4.16438922228 + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{if}\;x \leq -3800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, -0.0424927283095952, x \cdot -5.843575199059173\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ 4.16438922228 (/ y (* x (* x x)))))))
   (if (<= x -3800000000.0)
     t_0
     (if (<= x 2.0)
       (fma
        x
        (fma y -0.0424927283095952 (* x -5.843575199059173))
        (* z -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.16438922228 + (y / (x * (x * x))));
	double tmp;
	if (x <= -3800000000.0) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = fma(x, fma(y, -0.0424927283095952, (x * -5.843575199059173)), (z * -0.0424927283095952));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.16438922228 + Float64(y / Float64(x * Float64(x * x)))))
	tmp = 0.0
	if (x <= -3800000000.0)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = fma(x, fma(y, -0.0424927283095952, Float64(x * -5.843575199059173)), Float64(z * -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.16438922228 + N[(y / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3800000000.0], t$95$0, If[LessEqual[x, 2.0], N[(x * N[(y * -0.0424927283095952 + N[(x * -5.843575199059173), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e9 or 2 < x

   1. Initial program 21.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \left(\frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}} + \frac{y}{{x}^{3}}\right)\right) - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{{x}^{3}}\right)\right)} \]

   5. Simplified 93.8%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 93.3

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

   8. Simplified 93.3%

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

   if -3.8e9 < 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 96.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\frac{-137519416416}{23533438303} \cdot x + \frac{-1000000000}{23533438303} \cdot y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-137519416416}{23533438303} \cdot x + \frac{-1000000000}{23533438303} \cdot y\right) + \frac{-1000000000}{23533438303} \cdot z} \]

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, \frac{-1000000000}{23533438303}, \frac{-137519416416}{23533438303} \cdot x\right)}, \frac{-1000000000}{23533438303} \cdot z\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, \frac{-1000000000}{23533438303}, \color{blue}{x \cdot \frac{-137519416416}{23533438303}}\right), \frac{-1000000000}{23533438303} \cdot z\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, \frac{-1000000000}{23533438303}, \color{blue}{x \cdot \frac{-137519416416}{23533438303}}\right), \frac{-1000000000}{23533438303} \cdot z\right) \]

      8. *-lowering-*.f64 96.5

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

   10. Simplified 96.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, -0.0424927283095952, x \cdot -5.843575199059173\right), -0.0424927283095952 \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3800000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, -0.0424927283095952, x \cdot -5.843575199059173\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;x \cdot \left(y \cdot -0.0424927283095952\right)\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-101}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;y \cdot \left(x \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.5)
   (* x 4.16438922228)
   (if (<= x -4e-153)
     (* x (* y -0.0424927283095952))
     (if (<= x 1.12e-101)
       (* z -0.0424927283095952)
       (if (<= x 2.0) (* y (* x -0.0424927283095952)) (* x 4.16438922228))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.5) {
		tmp = x * 4.16438922228;
	} else if (x <= -4e-153) {
		tmp = x * (y * -0.0424927283095952);
	} else if (x <= 1.12e-101) {
		tmp = z * -0.0424927283095952;
	} else if (x <= 2.0) {
		tmp = y * (x * -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 <= (-5.5d0)) then
        tmp = x * 4.16438922228d0
    else if (x <= (-4d-153)) then
        tmp = x * (y * (-0.0424927283095952d0))
    else if (x <= 1.12d-101) then
        tmp = z * (-0.0424927283095952d0)
    else if (x <= 2.0d0) then
        tmp = y * (x * (-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 <= -5.5) {
		tmp = x * 4.16438922228;
	} else if (x <= -4e-153) {
		tmp = x * (y * -0.0424927283095952);
	} else if (x <= 1.12e-101) {
		tmp = z * -0.0424927283095952;
	} else if (x <= 2.0) {
		tmp = y * (x * -0.0424927283095952);
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.5:
		tmp = x * 4.16438922228
	elif x <= -4e-153:
		tmp = x * (y * -0.0424927283095952)
	elif x <= 1.12e-101:
		tmp = z * -0.0424927283095952
	elif x <= 2.0:
		tmp = y * (x * -0.0424927283095952)
	else:
		tmp = x * 4.16438922228
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.5)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= -4e-153)
		tmp = Float64(x * Float64(y * -0.0424927283095952));
	elseif (x <= 1.12e-101)
		tmp = Float64(z * -0.0424927283095952);
	elseif (x <= 2.0)
		tmp = Float64(y * Float64(x * -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 <= -5.5)
		tmp = x * 4.16438922228;
	elseif (x <= -4e-153)
		tmp = x * (y * -0.0424927283095952);
	elseif (x <= 1.12e-101)
		tmp = z * -0.0424927283095952;
	elseif (x <= 2.0)
		tmp = y * (x * -0.0424927283095952);
	else
		tmp = x * 4.16438922228;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.5], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, -4e-153], N[(x * N[(y * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.12e-101], N[(z * -0.0424927283095952), $MachinePrecision], If[LessEqual[x, 2.0], N[(y * N[(x * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.5 or 2 < x

   1. Initial program 23.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 84.6

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

   5. Simplified 84.6%

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

   if -5.5 < x < -4.00000000000000016e-153

   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 y around inf

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

      1. *-lowering-*.f64 63.2

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

   5. Simplified 63.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(x \cdot y\right)}{\color{blue}{\frac{23533438303}{500000000}}} \]
   7. Step-by-step derivation

   8. Simplified 62.6%

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(x \cdot y\right)}{\color{blue}{47.066876606}} \]

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

       5. associate-*r* N/A

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

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

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 62.5

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

   11. Simplified 62.5%

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

   if -4.00000000000000016e-153 < x < 1.12000000000000005e-101

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 74.9

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

   5. Simplified 74.9%

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

   if 1.12000000000000005e-101 < x < 2

   1. Initial program 99.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.1%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.1%

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

   8. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 80.0

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

   10. Simplified 80.0%

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

Alternative 9: 73.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y \cdot -0.0424927283095952\right)\\ \mathbf{if}\;x \leq -0.195:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-93}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (* y -0.0424927283095952))))
   (if (<= x -0.195)
     (* x 4.16438922228)
     (if (<= x -4e-153)
       t_0
       (if (<= x 2.4e-93)
         (* z -0.0424927283095952)
         (if (<= x 2.0) t_0 (* x 4.16438922228)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y * -0.0424927283095952);
	double tmp;
	if (x <= -0.195) {
		tmp = x * 4.16438922228;
	} else if (x <= -4e-153) {
		tmp = t_0;
	} else if (x <= 2.4e-93) {
		tmp = z * -0.0424927283095952;
	} else if (x <= 2.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = x * (y * (-0.0424927283095952d0))
    if (x <= (-0.195d0)) then
        tmp = x * 4.16438922228d0
    else if (x <= (-4d-153)) then
        tmp = t_0
    else if (x <= 2.4d-93) then
        tmp = z * (-0.0424927283095952d0)
    else if (x <= 2.0d0) then
        tmp = t_0
    else
        tmp = x * 4.16438922228d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y * -0.0424927283095952);
	double tmp;
	if (x <= -0.195) {
		tmp = x * 4.16438922228;
	} else if (x <= -4e-153) {
		tmp = t_0;
	} else if (x <= 2.4e-93) {
		tmp = z * -0.0424927283095952;
	} else if (x <= 2.0) {
		tmp = t_0;
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y * -0.0424927283095952)
	tmp = 0
	if x <= -0.195:
		tmp = x * 4.16438922228
	elif x <= -4e-153:
		tmp = t_0
	elif x <= 2.4e-93:
		tmp = z * -0.0424927283095952
	elif x <= 2.0:
		tmp = t_0
	else:
		tmp = x * 4.16438922228
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y * -0.0424927283095952))
	tmp = 0.0
	if (x <= -0.195)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= -4e-153)
		tmp = t_0;
	elseif (x <= 2.4e-93)
		tmp = Float64(z * -0.0424927283095952);
	elseif (x <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(x * 4.16438922228);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y * -0.0424927283095952);
	tmp = 0.0;
	if (x <= -0.195)
		tmp = x * 4.16438922228;
	elseif (x <= -4e-153)
		tmp = t_0;
	elseif (x <= 2.4e-93)
		tmp = z * -0.0424927283095952;
	elseif (x <= 2.0)
		tmp = t_0;
	else
		tmp = x * 4.16438922228;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y * -0.0424927283095952), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.195], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, -4e-153], t$95$0, If[LessEqual[x, 2.4e-93], N[(z * -0.0424927283095952), $MachinePrecision], If[LessEqual[x, 2.0], t$95$0, N[(x * 4.16438922228), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.19500000000000001 or 2 < x

   1. Initial program 23.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 84.6

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

   5. Simplified 84.6%

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

   if -0.19500000000000001 < x < -4.00000000000000016e-153 or 2.4000000000000001e-93 < 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 y around inf

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

      1. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(x \cdot y\right)}{\color{blue}{\frac{23533438303}{500000000}}} \]
   7. Step-by-step derivation

   8. Simplified 66.7%

      \[\leadsto \frac{\left(x - 2\right) \cdot \left(x \cdot y\right)}{\color{blue}{47.066876606}} \]

   9. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. *-commutative N/A

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

       4. metadata-eval N/A

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

       5. associate-*r* N/A

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

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

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

       7. associate-*r* N/A

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

       8. metadata-eval N/A

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

       9. *-commutative N/A

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

       10. *-lowering-*.f64 66.5

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

   11. Simplified 66.5%

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

   if -4.00000000000000016e-153 < x < 2.4000000000000001e-93

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 74.9

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

   5. Simplified 74.9%

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

Alternative 10: 73.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.0424927283095952 \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \leq -5.5:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-94}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.0424927283095952 (* x y))))
   (if (<= x -5.5)
     (* x 4.16438922228)
     (if (<= x -4e-153)
       t_0
       (if (<= x 2.15e-94)
         (* z -0.0424927283095952)
         (if (<= x 2.0) t_0 (* x 4.16438922228)))))))

C

double code(double x, double y, double z) {
	double t_0 = -0.0424927283095952 * (x * y);
	double tmp;
	if (x <= -5.5) {
		tmp = x * 4.16438922228;
	} else if (x <= -4e-153) {
		tmp = t_0;
	} else if (x <= 2.15e-94) {
		tmp = z * -0.0424927283095952;
	} else if (x <= 2.0) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (-0.0424927283095952d0) * (x * y)
    if (x <= (-5.5d0)) then
        tmp = x * 4.16438922228d0
    else if (x <= (-4d-153)) then
        tmp = t_0
    else if (x <= 2.15d-94) then
        tmp = z * (-0.0424927283095952d0)
    else if (x <= 2.0d0) then
        tmp = t_0
    else
        tmp = x * 4.16438922228d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.0424927283095952 * (x * y);
	double tmp;
	if (x <= -5.5) {
		tmp = x * 4.16438922228;
	} else if (x <= -4e-153) {
		tmp = t_0;
	} else if (x <= 2.15e-94) {
		tmp = z * -0.0424927283095952;
	} else if (x <= 2.0) {
		tmp = t_0;
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.0424927283095952 * (x * y)
	tmp = 0
	if x <= -5.5:
		tmp = x * 4.16438922228
	elif x <= -4e-153:
		tmp = t_0
	elif x <= 2.15e-94:
		tmp = z * -0.0424927283095952
	elif x <= 2.0:
		tmp = t_0
	else:
		tmp = x * 4.16438922228
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.0424927283095952 * Float64(x * y))
	tmp = 0.0
	if (x <= -5.5)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= -4e-153)
		tmp = t_0;
	elseif (x <= 2.15e-94)
		tmp = Float64(z * -0.0424927283095952);
	elseif (x <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(x * 4.16438922228);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.0424927283095952 * (x * y);
	tmp = 0.0;
	if (x <= -5.5)
		tmp = x * 4.16438922228;
	elseif (x <= -4e-153)
		tmp = t_0;
	elseif (x <= 2.15e-94)
		tmp = z * -0.0424927283095952;
	elseif (x <= 2.0)
		tmp = t_0;
	else
		tmp = x * 4.16438922228;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.0424927283095952 * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, -4e-153], t$95$0, If[LessEqual[x, 2.15e-94], N[(z * -0.0424927283095952), $MachinePrecision], If[LessEqual[x, 2.0], t$95$0, N[(x * 4.16438922228), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5 or 2 < x

   1. Initial program 23.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 84.6

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

   5. Simplified 84.6%

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

   if -5.5 < x < -4.00000000000000016e-153 or 2.1499999999999999e-94 < 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 y around inf

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

      1. *-lowering-*.f64 67.2

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

   5. Simplified 67.2%

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

   6. Taylor expanded in x around 0

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

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

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

      2. *-lowering-*.f64 66.3

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

   8. Simplified 66.3%

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

   if -4.00000000000000016e-153 < x < 2.1499999999999999e-94

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 74.9

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

   5. Simplified 74.9%

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

Alternative 11: 91.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{if}\;x \leq -3800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, -0.0424927283095952, x \cdot -5.843575199059173\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ 4.16438922228 (/ -110.1139242984811 x)))))
   (if (<= x -3800000000.0)
     t_0
     (if (<= x 2.0)
       (fma
        x
        (fma y -0.0424927283095952 (* x -5.843575199059173))
        (* z -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.16438922228 + (-110.1139242984811 / x));
	double tmp;
	if (x <= -3800000000.0) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = fma(x, fma(y, -0.0424927283095952, (x * -5.843575199059173)), (z * -0.0424927283095952));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)))
	tmp = 0.0
	if (x <= -3800000000.0)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = fma(x, fma(y, -0.0424927283095952, Float64(x * -5.843575199059173)), Float64(z * -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3800000000.0], t$95$0, If[LessEqual[x, 2.0], N[(x * N[(y * -0.0424927283095952 + N[(x * -5.843575199059173), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e9 or 2 < x

   1. Initial program 21.3%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} + \frac{104109730557}{25000000000}\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} - \frac{104109730557}{25000000000}\right)\right)} \]

      5. neg-sub0 N/A

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

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

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

      7. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} - \frac{104109730557}{25000000000}\right)\right)} \]

      8. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right) + \frac{104109730557}{25000000000}\right)} \]

      9. neg-sub0 N/A

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

      10. +-commutative N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

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

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

      16. metadata-eval 86.9

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

   5. Simplified 86.9%

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

   if -3.8e9 < 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 96.5%

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

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\frac{-137519416416}{23533438303} \cdot x + \frac{-1000000000}{23533438303} \cdot y\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-137519416416}{23533438303} \cdot x + \frac{-1000000000}{23533438303} \cdot y\right) + \frac{-1000000000}{23533438303} \cdot z} \]

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, \frac{-1000000000}{23533438303}, \frac{-137519416416}{23533438303} \cdot x\right)}, \frac{-1000000000}{23533438303} \cdot z\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, \frac{-1000000000}{23533438303}, \color{blue}{x \cdot \frac{-137519416416}{23533438303}}\right), \frac{-1000000000}{23533438303} \cdot z\right) \]

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

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(y, \frac{-1000000000}{23533438303}, \color{blue}{x \cdot \frac{-137519416416}{23533438303}}\right), \frac{-1000000000}{23533438303} \cdot z\right) \]

      8. *-lowering-*.f64 96.5

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

   10. Simplified 96.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(y, -0.0424927283095952, x \cdot -5.843575199059173\right), -0.0424927283095952 \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3800000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, -0.0424927283095952, x \cdot -5.843575199059173\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{if}\;x \leq -3800000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;-0.0424927283095952 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ 4.16438922228 (/ -110.1139242984811 x)))))
   (if (<= x -3800000000.0)
     t_0
     (if (<= x 2.0)
       (* -0.0424927283095952 (fma x (fma x 137.519416416 y) z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.16438922228 + (-110.1139242984811 / x));
	double tmp;
	if (x <= -3800000000.0) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = -0.0424927283095952 * fma(x, fma(x, 137.519416416, y), z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)))
	tmp = 0.0
	if (x <= -3800000000.0)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = Float64(-0.0424927283095952 * fma(x, fma(x, 137.519416416, y), z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3800000000.0], t$95$0, If[LessEqual[x, 2.0], N[(-0.0424927283095952 * N[(x * N[(x * 137.519416416 + y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e9 or 2 < x

   1. Initial program 21.3%

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

   3. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. neg-sub0 N/A

         \[\leadsto x \cdot \left(\color{blue}{\left(0 - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} + \frac{104109730557}{25000000000}\right) \]

      4. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} - \frac{104109730557}{25000000000}\right)\right)} \]

      5. neg-sub0 N/A

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

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

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

      7. neg-sub0 N/A

         \[\leadsto x \cdot \color{blue}{\left(0 - \left(\frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x} - \frac{104109730557}{25000000000}\right)\right)} \]

      8. associate-+l- N/A

         \[\leadsto x \cdot \color{blue}{\left(\left(0 - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right) + \frac{104109730557}{25000000000}\right)} \]

      9. neg-sub0 N/A

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

      10. +-commutative N/A

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

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

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. distribute-neg-frac N/A

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

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

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

      16. metadata-eval 86.9

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

   5. Simplified 86.9%

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

   if -3.8e9 < 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 96.5%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 96.5%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{137.519416416}, y\right), z\right) \cdot -0.0424927283095952 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3800000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;-0.0424927283095952 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3800000000:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;-0.0424927283095952 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3800000000.0)
   (* x 4.16438922228)
   (if (<= x 2.0)
     (* -0.0424927283095952 (fma x (fma x 137.519416416 y) z))
     (* x 4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3800000000.0) {
		tmp = x * 4.16438922228;
	} else if (x <= 2.0) {
		tmp = -0.0424927283095952 * fma(x, fma(x, 137.519416416, y), z);
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3800000000.0)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= 2.0)
		tmp = Float64(-0.0424927283095952 * fma(x, fma(x, 137.519416416, y), z));
	else
		tmp = Float64(x * 4.16438922228);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3800000000.0], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, 2.0], N[(-0.0424927283095952 * N[(x * N[(x * 137.519416416 + y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e9 or 2 < x

   1. Initial program 21.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 86.5

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

   5. Simplified 86.5%

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

   if -3.8e9 < 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 96.5%

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

   8. Taylor expanded in x around 0

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

   10. Simplified 96.5%

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{137.519416416}, y\right), z\right) \cdot -0.0424927283095952 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3800000000:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;-0.0424927283095952 \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3800000000:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;-0.0424927283095952 \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3800000000.0)
   (* x 4.16438922228)
   (if (<= x 2.0) (* -0.0424927283095952 (fma y x z)) (* x 4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3800000000.0) {
		tmp = x * 4.16438922228;
	} else if (x <= 2.0) {
		tmp = -0.0424927283095952 * fma(y, x, z);
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3800000000.0)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= 2.0)
		tmp = Float64(-0.0424927283095952 * fma(y, x, z));
	else
		tmp = Float64(x * 4.16438922228);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3800000000.0], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, 2.0], N[(-0.0424927283095952 * N[(y * x + z), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e9 or 2 < x

   1. Initial program 21.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 86.5

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

   5. Simplified 86.5%

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

   if -3.8e9 < 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 96.5%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 96.3

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

   10. Simplified 96.3%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3800000000:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;-0.0424927283095952 \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3800000000:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-53}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3800000000.0)
   (* x 4.16438922228)
   (if (<= x 1.45e-53) (* z -0.0424927283095952) (* x 4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3800000000.0) {
		tmp = x * 4.16438922228;
	} else if (x <= 1.45e-53) {
		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 <= (-3800000000.0d0)) then
        tmp = x * 4.16438922228d0
    else if (x <= 1.45d-53) 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 <= -3800000000.0) {
		tmp = x * 4.16438922228;
	} else if (x <= 1.45e-53) {
		tmp = z * -0.0424927283095952;
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3800000000.0:
		tmp = x * 4.16438922228
	elif x <= 1.45e-53:
		tmp = z * -0.0424927283095952
	else:
		tmp = x * 4.16438922228
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3800000000.0)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= 1.45e-53)
		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 <= -3800000000.0)
		tmp = x * 4.16438922228;
	elseif (x <= 1.45e-53)
		tmp = z * -0.0424927283095952;
	else
		tmp = x * 4.16438922228;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3800000000.0], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, 1.45e-53], N[(z * -0.0424927283095952), $MachinePrecision], N[(x * 4.16438922228), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8e9 or 1.4499999999999999e-53 < x

   1. Initial program 23.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 84.6

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

   5. Simplified 84.6%

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

   if -3.8e9 < x < 1.4499999999999999e-53

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 60.7

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

   5. Simplified 60.7%

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

Alternative 16: 44.1% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ x \cdot 4.16438922228 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x 4.16438922228))

C

double code(double x, double y, double z) {
	return x * 4.16438922228;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x * 4.16438922228d0
end function

Java

public static double code(double x, double y, double z) {
	return x * 4.16438922228;
}

Python

def code(x, y, z):
	return x * 4.16438922228

Julia

function code(x, y, z)
	return Float64(x * 4.16438922228)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * 4.16438922228;
end

Wolfram

code[x_, y_, z_] := N[(x * 4.16438922228), $MachinePrecision]

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 46.2

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

5. Simplified 46.2%

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

Alternative 17: 2.3% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ x \cdot -50.89257292696055 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* x -50.89257292696055))

C

double code(double x, double y, double z) {
	return x * -50.89257292696055;
}

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 * (-50.89257292696055d0)
end function

Java

public static double code(double x, double y, double z) {
	return x * -50.89257292696055;
}

Python

def code(x, y, z):
	return x * -50.89257292696055

Julia

function code(x, y, z)
	return Float64(x * -50.89257292696055)
end

MATLAB

function tmp = code(x, y, z)
	tmp = x * -50.89257292696055;
end

Wolfram

code[x_, y_, z_] := N[(x * -50.89257292696055), $MachinePrecision]

Derivation

1. Initial program 59.2%

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

   1. associate-/l* N/A

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

   2. flip-- N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

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

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

4. Applied egg-rr 62.5%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{104109730557}{25000000000} - \frac{12723143231740136880149}{125000000000000000000} \cdot \frac{1}{x}\right)\right)} \cdot \frac{1}{x + 2} \]
6. Step-by-step derivation

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

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{104109730557}{25000000000} - \frac{12723143231740136880149}{125000000000000000000} \cdot \frac{1}{x}\right)\right)} \cdot \frac{1}{x + 2} \]

   2. unpow2 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{104109730557}{25000000000} - \frac{12723143231740136880149}{125000000000000000000} \cdot \frac{1}{x}\right)\right) \cdot \frac{1}{x + 2} \]

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

      \[\leadsto \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{104109730557}{25000000000} - \frac{12723143231740136880149}{125000000000000000000} \cdot \frac{1}{x}\right)\right) \cdot \frac{1}{x + 2} \]

   4. sub-neg N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{104109730557}{25000000000} + \left(\mathsf{neg}\left(\frac{12723143231740136880149}{125000000000000000000} \cdot \frac{1}{x}\right)\right)\right)}\right) \cdot \frac{1}{x + 2} \]

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

      \[\leadsto \left(\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{104109730557}{25000000000} + \left(\mathsf{neg}\left(\frac{12723143231740136880149}{125000000000000000000} \cdot \frac{1}{x}\right)\right)\right)}\right) \cdot \frac{1}{x + 2} \]

   6. associate-*r/ N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{104109730557}{25000000000} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{12723143231740136880149}{125000000000000000000} \cdot 1}{x}}\right)\right)\right)\right) \cdot \frac{1}{x + 2} \]

   7. metadata-eval N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{104109730557}{25000000000} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{12723143231740136880149}{125000000000000000000}}}{x}\right)\right)\right)\right) \cdot \frac{1}{x + 2} \]

   8. distribute-neg-frac N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{104109730557}{25000000000} + \color{blue}{\frac{\mathsf{neg}\left(\frac{12723143231740136880149}{125000000000000000000}\right)}{x}}\right)\right) \cdot \frac{1}{x + 2} \]

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

      \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\frac{104109730557}{25000000000} + \color{blue}{\frac{\mathsf{neg}\left(\frac{12723143231740136880149}{125000000000000000000}\right)}{x}}\right)\right) \cdot \frac{1}{x + 2} \]

   10. metadata-eval 23.8

       \[\leadsto \left(\left(x \cdot x\right) \cdot \left(4.16438922228 + \frac{\color{blue}{-101.7851458539211}}{x}\right)\right) \cdot \frac{1}{x + 2} \]

7. Simplified 23.8%

   \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(4.16438922228 + \frac{-101.7851458539211}{x}\right)\right)} \cdot \frac{1}{x + 2} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{-12723143231740136880149}{250000000000000000000} \cdot x} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{-12723143231740136880149}{250000000000000000000}} \]

   2. *-lowering-*.f64 2.2

      \[\leadsto \color{blue}{x \cdot -50.89257292696055} \]

10. Simplified 2.2%

    \[\leadsto \color{blue}{x \cdot -50.89257292696055} \]
11. Add Preprocessing

Developer Target 1: 98.9% 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 2024205 
(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)))