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

Percentage Accurate: 59.1% → 98.6%

Time: 18.1s

Alternatives: 19

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: 59.1% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+303}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), x \cdot x, \mathsf{fma}\left(x, 313.399215894, 47.066876606\right)\right)}\right) \cdot \frac{1}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          x
          (+
           (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
           y))
         z))
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606))
      1e+303)
   (*
    (*
     (fma x x -4.0)
     (/
      (fma
       x
       (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
       z)
      (fma
       (fma x (+ x 43.3400022514) 263.505074721)
       (* x x)
       (fma x 313.399215894 47.066876606))))
    (/ 1.0 (+ x 2.0)))
   (fma x 4.16438922228 (/ y (* x x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+303) {
		tmp = (fma(x, x, -4.0) * (fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(fma(x, (x + 43.3400022514), 263.505074721), (x * x), fma(x, 313.399215894, 47.066876606)))) * (1.0 / (x + 2.0));
	} else {
		tmp = fma(x, 4.16438922228, (y / (x * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+303)
		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(fma(x, Float64(x + 43.3400022514), 263.505074721), Float64(x * x), fma(x, 313.399215894, 47.066876606)))) * Float64(1.0 / Float64(x + 2.0)));
	else
		tmp = fma(x, 4.16438922228, Float64(y / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], 1e+303], 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[(N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] * N[(x * x), $MachinePrecision] + N[(x * 313.399215894 + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64))) < 1e303

   1. Initial program 97.7%

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

      1. associate-/l* N/A

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

      2. flip-- N/A

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

      3. 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 99.0%

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

      1. distribute-rgt-in N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. accelerator-lowering-fma.f64 99.0

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

   6. Applied egg-rr 99.0%

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

   if 1e303 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.2%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}} - \frac{-104109730557}{25000000000} \cdot x} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 99.2

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

   11. Simplified 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+303}:\\ \;\;\;\;\left(\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), x \cdot x, \mathsf{fma}\left(x, 313.399215894, 47.066876606\right)\right)}\right) \cdot \frac{1}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+303}:\\ \;\;\;\;\frac{1}{x + 2} \cdot \left(\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          x
          (+
           (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
           y))
         z))
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606))
      1e+303)
   (*
    (/ 1.0 (+ x 2.0))
    (*
     (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))))
   (fma x 4.16438922228 (/ y (* x x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+303) {
		tmp = (1.0 / (x + 2.0)) * (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)));
	} else {
		tmp = fma(x, 4.16438922228, (y / (x * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+303)
		tmp = Float64(Float64(1.0 / Float64(x + 2.0)) * 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))));
	else
		tmp = fma(x, 4.16438922228, Float64(y / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], 1e+303], N[(N[(1.0 / N[(x + 2.0), $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision], N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64))) < 1e303

   1. Initial program 97.7%

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

      1. associate-/l* N/A

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

      2. flip-- N/A

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

      3. 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 99.0%

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

   if 1e303 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.2%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}} - \frac{-104109730557}{25000000000} \cdot x} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 99.2

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

   11. Simplified 99.2%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+303}:\\ \;\;\;\;\frac{1}{x + 2} \cdot \left(\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          x
          (+
           (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
           y))
         z))
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606))
      1e+303)
   (*
    (fma
     x
     (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
     z)
    (/
     (+ x -2.0)
     (fma
      x
      (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
      47.066876606)))
   (fma x 4.16438922228 (/ y (* x x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+303) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * ((x + -2.0) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606));
	} else {
		tmp = fma(x, 4.16438922228, (y / (x * x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+303)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * Float64(Float64(x + -2.0) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)));
	else
		tmp = fma(x, 4.16438922228, Float64(y / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], 1e+303], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * N[(N[(x + -2.0), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64))) < 1e303

   1. Initial program 97.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 98.7%

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

   if 1e303 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.2%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}} - \frac{-104109730557}{25000000000} \cdot x} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 99.2

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

   11. Simplified 99.2%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+303}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 67000000000000:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y - 130977.50649958357}{x} - -3655.1204654076414}{x} - 110.1139242984811}{x} - x \cdot -4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2e+15)
   (fma x 4.16438922228 (/ y (* x x)))
   (if (<= x 67000000000000.0)
     (/
      (* (- x 2.0) (fma x (fma x 137.519416416 y) z))
      (+
       (*
        x
        (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
       47.066876606))
     (-
      (*
       x
       (/
        (-
         (/ (- (/ (- y 130977.50649958357) x) -3655.1204654076414) x)
         110.1139242984811)
        x))
      (* x -4.16438922228)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2e+15) {
		tmp = fma(x, 4.16438922228, (y / (x * x)));
	} else if (x <= 67000000000000.0) {
		tmp = ((x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606);
	} else {
		tmp = (x * ((((((y - 130977.50649958357) / x) - -3655.1204654076414) / x) - 110.1139242984811) / x)) - (x * -4.16438922228);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2e+15)
		tmp = fma(x, 4.16438922228, Float64(y / Float64(x * x)));
	elseif (x <= 67000000000000.0)
		tmp = Float64(Float64(Float64(x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606));
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(y - 130977.50649958357) / x) - -3655.1204654076414) / x) - 110.1139242984811) / x)) - Float64(x * -4.16438922228));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2e+15], N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 67000000000000.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(x * N[(x * 137.519416416 + y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[(N[(N[(N[(y - 130977.50649958357), $MachinePrecision] / x), $MachinePrecision] - -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - N[(x * -4.16438922228), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e15

   1. Initial program 14.9%

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

   3. 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 99.3%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 99.3

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

   8. Simplified 99.3%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}} - \frac{-104109730557}{25000000000} \cdot x} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 99.3

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

   11. Simplified 99.3%

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

   if -2e15 < x < 6.7e13

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 99.2

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

   5. Simplified 99.2%

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

   if 6.7e13 < x

   1. Initial program 18.0%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.3%

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

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

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

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

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

      6. distribute-neg-frac N/A

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

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

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

   7. Applied egg-rr 97.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 67000000000000:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y - 130977.50649958357}{x} - -3655.1204654076414}{x} - 110.1139242984811}{x} - x \cdot -4.16438922228\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 31:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, 4.16438922228, \mathsf{fma}\left(x, 78.6994924154, 137.519416416\right)\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 10.238818846568002, -1.787568985856513\right), 0.3041881842569256\right), -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y - 130977.50649958357}{x} - -3655.1204654076414}{x} - 110.1139242984811}{x} - x \cdot -4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.175)
   (fma x 4.16438922228 (/ y (* x x)))
   (if (<= x 31.0)
     (*
      (fma
       x
       (fma
        x
        (fma (* x x) 4.16438922228 (fma x 78.6994924154 137.519416416))
        y)
       z)
      (fma
       x
       (fma x (fma x 10.238818846568002 -1.787568985856513) 0.3041881842569256)
       -0.0424927283095952))
     (-
      (*
       x
       (/
        (-
         (/ (- (/ (- y 130977.50649958357) x) -3655.1204654076414) x)
         110.1139242984811)
        x))
      (* x -4.16438922228)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.175) {
		tmp = fma(x, 4.16438922228, (y / (x * x)));
	} else if (x <= 31.0) {
		tmp = fma(x, fma(x, fma((x * x), 4.16438922228, fma(x, 78.6994924154, 137.519416416)), y), z) * fma(x, fma(x, fma(x, 10.238818846568002, -1.787568985856513), 0.3041881842569256), -0.0424927283095952);
	} else {
		tmp = (x * ((((((y - 130977.50649958357) / x) - -3655.1204654076414) / x) - 110.1139242984811) / x)) - (x * -4.16438922228);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.175)
		tmp = fma(x, 4.16438922228, Float64(y / Float64(x * x)));
	elseif (x <= 31.0)
		tmp = Float64(fma(x, fma(x, fma(Float64(x * x), 4.16438922228, fma(x, 78.6994924154, 137.519416416)), y), z) * fma(x, fma(x, fma(x, 10.238818846568002, -1.787568985856513), 0.3041881842569256), -0.0424927283095952));
	else
		tmp = Float64(Float64(x * Float64(Float64(Float64(Float64(Float64(Float64(y - 130977.50649958357) / x) - -3655.1204654076414) / x) - 110.1139242984811) / x)) - Float64(x * -4.16438922228));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.175], N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 31.0], N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 4.16438922228 + N[(x * 78.6994924154 + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * N[(x * N[(x * N[(x * 10.238818846568002 + -1.787568985856513), $MachinePrecision] + 0.3041881842569256), $MachinePrecision] + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(N[(N[(N[(N[(y - 130977.50649958357), $MachinePrecision] / x), $MachinePrecision] - -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] - N[(x * -4.16438922228), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.17499999999999999

   1. Initial program 16.2%

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

   3. Taylor expanded in 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%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 99.0

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

   8. Simplified 99.0%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}} - \frac{-104109730557}{25000000000} \cdot x} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 99.1

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

   11. Simplified 99.1%

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

   if -0.17499999999999999 < x < 31

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 99.3

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

   7. Simplified 99.3%

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

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 99.3

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

   9. Applied egg-rr 99.3%

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

   if 31 < x

   1. Initial program 19.5%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 96.3%

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

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

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

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

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

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

      6. distribute-neg-frac N/A

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

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

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

   7. Applied egg-rr 96.4%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 31:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, 4.16438922228, \mathsf{fma}\left(x, 78.6994924154, 137.519416416\right)\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 10.238818846568002, -1.787568985856513\right), 0.3041881842569256\right), -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y - 130977.50649958357}{x} - -3655.1204654076414}{x} - 110.1139242984811}{x} - x \cdot -4.16438922228\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.155:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, 4.16438922228, \mathsf{fma}\left(x, 78.6994924154, 137.519416416\right)\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 10.238818846568002, -1.787568985856513\right), 0.3041881842569256\right), -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x 4.16438922228 (/ y (* x x)))))
   (if (<= x -0.175)
     t_0
     (if (<= x 0.155)
       (*
        (fma
         x
         (fma
          x
          (fma (* x x) 4.16438922228 (fma x 78.6994924154 137.519416416))
          y)
         z)
        (fma
         x
         (fma
          x
          (fma x 10.238818846568002 -1.787568985856513)
          0.3041881842569256)
         -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, 4.16438922228, (y / (x * x)));
	double tmp;
	if (x <= -0.175) {
		tmp = t_0;
	} else if (x <= 0.155) {
		tmp = fma(x, fma(x, fma((x * x), 4.16438922228, fma(x, 78.6994924154, 137.519416416)), y), z) * fma(x, fma(x, fma(x, 10.238818846568002, -1.787568985856513), 0.3041881842569256), -0.0424927283095952);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, 4.16438922228, Float64(y / Float64(x * x)))
	tmp = 0.0
	if (x <= -0.175)
		tmp = t_0;
	elseif (x <= 0.155)
		tmp = Float64(fma(x, fma(x, fma(Float64(x * x), 4.16438922228, fma(x, 78.6994924154, 137.519416416)), y), z) * fma(x, fma(x, fma(x, 10.238818846568002, -1.787568985856513), 0.3041881842569256), -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.175], t$95$0, If[LessEqual[x, 0.155], N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 4.16438922228 + N[(x * 78.6994924154 + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * N[(x * N[(x * N[(x * 10.238818846568002 + -1.787568985856513), $MachinePrecision] + 0.3041881842569256), $MachinePrecision] + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.17499999999999999 or 0.154999999999999999 < x

   1. Initial program 17.7%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.8%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 97.8

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

   8. Simplified 97.8%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}} - \frac{-104109730557}{25000000000} \cdot x} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 97.8

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

   11. Simplified 97.8%

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

   if -0.17499999999999999 < x < 0.154999999999999999

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 99.3

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

   7. Simplified 99.3%

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

      1. distribute-lft-in N/A

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

      2. associate-+l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. accelerator-lowering-fma.f64 99.3

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

   9. Applied egg-rr 99.3%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x \cdot x, 4.16438922228, \mathsf{fma}\left(x, 78.6994924154, 137.519416416\right)\right)}, y\right), z\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 10.238818846568002, -1.787568985856513\right), 0.3041881842569256\right), -0.0424927283095952\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 95.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 10.5:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 10.238818846568002, -1.787568985856513\right), 0.3041881842569256\right), -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x 4.16438922228 (/ y (* x x)))))
   (if (<= x -5.5)
     t_0
     (if (<= x 10.5)
       (*
        (fma
         x
         (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
         z)
        (fma
         x
         (fma
          x
          (fma x 10.238818846568002 -1.787568985856513)
          0.3041881842569256)
         -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, 4.16438922228, (y / (x * x)));
	double tmp;
	if (x <= -5.5) {
		tmp = t_0;
	} else if (x <= 10.5) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, fma(x, fma(x, 10.238818846568002, -1.787568985856513), 0.3041881842569256), -0.0424927283095952);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, 4.16438922228, Float64(y / Float64(x * x)))
	tmp = 0.0
	if (x <= -5.5)
		tmp = t_0;
	elseif (x <= 10.5)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, fma(x, fma(x, 10.238818846568002, -1.787568985856513), 0.3041881842569256), -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5], t$95$0, If[LessEqual[x, 10.5], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * N[(x * N[(x * N[(x * 10.238818846568002 + -1.787568985856513), $MachinePrecision] + 0.3041881842569256), $MachinePrecision] + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5 or 10.5 < x

   1. Initial program 17.7%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.8%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 97.8

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

   8. Simplified 97.8%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}} - \frac{-104109730557}{25000000000} \cdot x} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 97.8

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

   11. Simplified 97.8%

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

   if -5.5 < x < 10.5

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. +-commutative N/A

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

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

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. metadata-eval N/A

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

      9. accelerator-lowering-fma.f64 99.3

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

   7. Simplified 99.3%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 10.238818846568002, -1.787568985856513\right), 0.3041881842569256\right), -0.0424927283095952\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 95.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.787568985856513, 0.3041881842569256\right), -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x 4.16438922228 (/ y (* x x)))))
   (if (<= x -5.5)
     t_0
     (if (<= x 2.0)
       (*
        (fma
         x
         (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
         z)
        (fma
         x
         (fma x -1.787568985856513 0.3041881842569256)
         -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, 4.16438922228, (y / (x * x)));
	double tmp;
	if (x <= -5.5) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, fma(x, -1.787568985856513, 0.3041881842569256), -0.0424927283095952);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, 4.16438922228, Float64(y / Float64(x * x)))
	tmp = 0.0
	if (x <= -5.5)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, fma(x, -1.787568985856513, 0.3041881842569256), -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5], t$95$0, If[LessEqual[x, 2.0], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * N[(x * N[(x * -1.787568985856513 + 0.3041881842569256), $MachinePrecision] + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5 or 2 < x

   1. Initial program 17.7%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.8%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 97.8

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

   8. Simplified 97.8%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}} - \frac{-104109730557}{25000000000} \cdot x} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 97.8

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

   11. Simplified 97.8%

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

   if -5.5 < x < 2

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 99.2

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

   7. Simplified 99.2%

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.787568985856513, 0.3041881842569256\right), -0.0424927283095952\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 95.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 22.5:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x 4.16438922228 (/ y (* x x)))))
   (if (<= x -5.5)
     t_0
     (if (<= x 22.5)
       (*
        (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 = fma(x, 4.16438922228, (y / (x * x)));
	double tmp;
	if (x <= -5.5) {
		tmp = t_0;
	} else if (x <= 22.5) {
		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 = fma(x, 4.16438922228, Float64(y / Float64(x * x)))
	tmp = 0.0
	if (x <= -5.5)
		tmp = t_0;
	elseif (x <= 22.5)
		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 * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5], t$95$0, If[LessEqual[x, 22.5], 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 < -5.5 or 22.5 < x

   1. Initial program 17.7%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.8%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 97.8

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

   8. Simplified 97.8%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}} - \frac{-104109730557}{25000000000} \cdot x} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 97.8

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

   11. Simplified 97.8%

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

   if -5.5 < x < 22.5

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 98.9

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

   7. Simplified 98.9%

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

Alternative 10: 94.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x 4.16438922228 (/ y (* x x)))))
   (if (<= x -5.5)
     t_0
     (if (<= x 2.0)
       (*
        (fma
         x
         (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
         z)
        -0.0424927283095952)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, 4.16438922228, (y / (x * x)));
	double tmp;
	if (x <= -5.5) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * -0.0424927283095952;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, 4.16438922228, Float64(y / Float64(x * x)))
	tmp = 0.0
	if (x <= -5.5)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * -0.0424927283095952);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5], t$95$0, If[LessEqual[x, 2.0], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * -0.0424927283095952), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5 or 2 < x

   1. Initial program 17.7%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.8%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 97.8

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

   8. Simplified 97.8%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}} - \frac{-104109730557}{25000000000} \cdot x} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 97.8

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

   11. Simplified 97.8%

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

   if -5.5 < x < 2

   1. Initial program 99.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 98.4%

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

Alternative 11: 79.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -1.15 \cdot 10^{-23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{z \cdot \left(x + -2\right)}{47.066876606}\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.3041881842569256, -5.843575199059173\right), y \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x 4.16438922228 (/ y (* x x)))))
   (if (<= x -1.15e-23)
     t_0
     (if (<= x 5.5e-83)
       (/ (* z (+ x -2.0)) 47.066876606)
       (if (<= x 2.0)
         (*
          x
          (fma
           x
           (fma y 0.3041881842569256 -5.843575199059173)
           (* y -0.0424927283095952)))
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, 4.16438922228, (y / (x * x)));
	double tmp;
	if (x <= -1.15e-23) {
		tmp = t_0;
	} else if (x <= 5.5e-83) {
		tmp = (z * (x + -2.0)) / 47.066876606;
	} else if (x <= 2.0) {
		tmp = x * fma(x, fma(y, 0.3041881842569256, -5.843575199059173), (y * -0.0424927283095952));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, 4.16438922228, Float64(y / Float64(x * x)))
	tmp = 0.0
	if (x <= -1.15e-23)
		tmp = t_0;
	elseif (x <= 5.5e-83)
		tmp = Float64(Float64(z * Float64(x + -2.0)) / 47.066876606);
	elseif (x <= 2.0)
		tmp = Float64(x * fma(x, fma(y, 0.3041881842569256, -5.843575199059173), Float64(y * -0.0424927283095952)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.15e-23], t$95$0, If[LessEqual[x, 5.5e-83], N[(N[(z * N[(x + -2.0), $MachinePrecision]), $MachinePrecision] / 47.066876606), $MachinePrecision], If[LessEqual[x, 2.0], N[(x * N[(x * N[(y * 0.3041881842569256 + -5.843575199059173), $MachinePrecision] + N[(y * -0.0424927283095952), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.15000000000000005e-23 or 2 < x

   1. Initial program 18.9%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 96.3%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 96.4

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

   8. Simplified 96.4%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}} - \frac{-104109730557}{25000000000} \cdot x} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 96.4

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

   11. Simplified 96.4%

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

   if -1.15000000000000005e-23 < x < 5.49999999999999964e-83

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 76.8

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

   5. Simplified 76.8%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 76.8%

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

   if 5.49999999999999964e-83 < 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. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) + x \cdot \left(\frac{500000000}{23533438303} \cdot \left(y - \frac{4297481763}{15625000}\right) - \left(\frac{-131752537360500000000}{553822718361107519809} \cdot z + \frac{156699607947}{23533438303} \cdot \left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right)\right)\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)} \]

   5. Simplified 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0212463641547976, \mathsf{fma}\left(y, -0.0424927283095952, z \cdot 0.28294182010212804\right)\right), -6.658593866711955, \mathsf{fma}\left(z, 0.23789659216289816, \mathsf{fma}\left(y, 0.0212463641547976, -5.843575199059173\right)\right)\right), z \cdot 0.28294182010212804\right)\right), z \cdot -0.0424927283095952\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y + x \cdot \left(\left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) - \frac{137519416416}{23533438303}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y + x \cdot \left(\left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) - \frac{137519416416}{23533438303}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) - \frac{137519416416}{23533438303}\right) + \frac{-1000000000}{23533438303} \cdot y\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) - \frac{137519416416}{23533438303}, \frac{-1000000000}{23533438303} \cdot y\right)} \]

      4. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) + \left(\mathsf{neg}\left(\frac{137519416416}{23533438303}\right)\right)}, \frac{-1000000000}{23533438303} \cdot y\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{y \cdot \left(\frac{500000000}{23533438303} + \frac{156699607947000000000}{553822718361107519809}\right)} + \left(\mathsf{neg}\left(\frac{137519416416}{23533438303}\right)\right), \frac{-1000000000}{23533438303} \cdot y\right) \]

      6. metadata-eval N/A

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

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

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, \frac{168466327098500000000}{553822718361107519809}, \mathsf{neg}\left(\frac{137519416416}{23533438303}\right)\right)}, \frac{-1000000000}{23533438303} \cdot y\right) \]

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 63.4

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

   8. Simplified 63.4%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.3041881842569256, -5.843575199059173\right), -0.0424927283095952 \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-83}:\\ \;\;\;\;\frac{z \cdot \left(x + -2\right)}{47.066876606}\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.3041881842569256, -5.843575199059173\right), y \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{0.24013125253755718}{x}}\\ \mathbf{if}\;x \leq -210:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-85}:\\ \;\;\;\;\frac{z \cdot \left(x + -2\right)}{47.066876606}\\ \mathbf{elif}\;x \leq 2450000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.3041881842569256, -5.843575199059173\right), y \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ 0.24013125253755718 x))))
   (if (<= x -210.0)
     t_0
     (if (<= x 1.15e-85)
       (/ (* z (+ x -2.0)) 47.066876606)
       (if (<= x 2450000.0)
         (*
          x
          (fma
           x
           (fma y 0.3041881842569256 -5.843575199059173)
           (* y -0.0424927283095952)))
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 / (0.24013125253755718 / x);
	double tmp;
	if (x <= -210.0) {
		tmp = t_0;
	} else if (x <= 1.15e-85) {
		tmp = (z * (x + -2.0)) / 47.066876606;
	} else if (x <= 2450000.0) {
		tmp = x * fma(x, fma(y, 0.3041881842569256, -5.843575199059173), (y * -0.0424927283095952));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(1.0 / Float64(0.24013125253755718 / x))
	tmp = 0.0
	if (x <= -210.0)
		tmp = t_0;
	elseif (x <= 1.15e-85)
		tmp = Float64(Float64(z * Float64(x + -2.0)) / 47.066876606);
	elseif (x <= 2450000.0)
		tmp = Float64(x * fma(x, fma(y, 0.3041881842569256, -5.843575199059173), Float64(y * -0.0424927283095952)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 / N[(0.24013125253755718 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -210.0], t$95$0, If[LessEqual[x, 1.15e-85], N[(N[(z * N[(x + -2.0), $MachinePrecision]), $MachinePrecision] / 47.066876606), $MachinePrecision], If[LessEqual[x, 2450000.0], N[(x * N[(x * N[(y * 0.3041881842569256 + -5.843575199059173), $MachinePrecision] + N[(y * -0.0424927283095952), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -210 or 2.45e6 < x

   1. Initial program 17.0%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 98.2%

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

   6. Taylor expanded in x around 0

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

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

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

      2. +-commutative N/A

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

      3. associate--l+ N/A

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

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

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

      5. +-commutative N/A

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

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

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

      7. sub-neg N/A

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

      8. *-commutative N/A

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

      9. metadata-eval N/A

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

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

          \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{-13764240537310136880149}{125000000000000000000}\right)}, \frac{2284450290879775841688574159837293}{625000000000000000000000000000}\right), y - \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000}\right)}{{x}^{2}} \]

      11. sub-neg N/A

          \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{-13764240537310136880149}{125000000000000000000}\right), \frac{2284450290879775841688574159837293}{625000000000000000000000000000}\right), \color{blue}{y + \left(\mathsf{neg}\left(\frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000}\right)\right)}\right)}{{x}^{2}} \]

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

          \[\leadsto \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{-13764240537310136880149}{125000000000000000000}\right), \frac{2284450290879775841688574159837293}{625000000000000000000000000000}\right), \color{blue}{y + \left(\mathsf{neg}\left(\frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000}\right)\right)}\right)}{{x}^{2}} \]

      13. metadata-eval N/A

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 30.0

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

   8. Simplified 30.0%

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

      1. clear-num N/A

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

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

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

      3. associate-/l* N/A

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

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

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

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

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

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

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

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

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

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

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

      11. accelerator-lowering-fma.f64 34.2

          \[\leadsto \frac{1}{x \cdot \frac{x}{-130977.50649958357 + \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)}, 3655.1204654076414\right), y\right)}} \]

   10. Applied egg-rr 34.2%

       \[\leadsto \color{blue}{\frac{1}{x \cdot \frac{x}{-130977.50649958357 + \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right), 3655.1204654076414\right), y\right)}}} \]

   11. Taylor expanded in x around inf

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

       1. /-lowering-/.f64 88.6

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

   13. Simplified 88.6%

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

   if -210 < x < 1.15e-85

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 75.5

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

   5. Simplified 75.5%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 75.5%

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

   if 1.15e-85 < x < 2.45e6

   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. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) + x \cdot \left(\frac{500000000}{23533438303} \cdot \left(y - \frac{4297481763}{15625000}\right) - \left(\frac{-131752537360500000000}{553822718361107519809} \cdot z + \frac{156699607947}{23533438303} \cdot \left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right)\right)\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)} \]

   5. Simplified 89.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0212463641547976, \mathsf{fma}\left(y, -0.0424927283095952, z \cdot 0.28294182010212804\right)\right), -6.658593866711955, \mathsf{fma}\left(z, 0.23789659216289816, \mathsf{fma}\left(y, 0.0212463641547976, -5.843575199059173\right)\right)\right), z \cdot 0.28294182010212804\right)\right), z \cdot -0.0424927283095952\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y + x \cdot \left(\left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) - \frac{137519416416}{23533438303}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y + x \cdot \left(\left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) - \frac{137519416416}{23533438303}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) - \frac{137519416416}{23533438303}\right) + \frac{-1000000000}{23533438303} \cdot y\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) - \frac{137519416416}{23533438303}, \frac{-1000000000}{23533438303} \cdot y\right)} \]

      4. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) + \left(\mathsf{neg}\left(\frac{137519416416}{23533438303}\right)\right)}, \frac{-1000000000}{23533438303} \cdot y\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{y \cdot \left(\frac{500000000}{23533438303} + \frac{156699607947000000000}{553822718361107519809}\right)} + \left(\mathsf{neg}\left(\frac{137519416416}{23533438303}\right)\right), \frac{-1000000000}{23533438303} \cdot y\right) \]

      6. metadata-eval N/A

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

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

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, \frac{168466327098500000000}{553822718361107519809}, \mathsf{neg}\left(\frac{137519416416}{23533438303}\right)\right)}, \frac{-1000000000}{23533438303} \cdot y\right) \]

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 60.0

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

   8. Simplified 60.0%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.3041881842569256, -5.843575199059173\right), -0.0424927283095952 \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -210:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-85}:\\ \;\;\;\;\frac{z \cdot \left(x + -2\right)}{47.066876606}\\ \mathbf{elif}\;x \leq 2450000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.3041881842569256, -5.843575199059173\right), y \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.009:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{z \cdot \left(x + -2\right)}{47.066876606}\\ \mathbf{elif}\;x \leq 2600000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.3041881842569256, -5.843575199059173\right), y \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.009)
   (* x 4.16438922228)
   (if (<= x 6.2e-85)
     (/ (* z (+ x -2.0)) 47.066876606)
     (if (<= x 2600000.0)
       (*
        x
        (fma
         x
         (fma y 0.3041881842569256 -5.843575199059173)
         (* y -0.0424927283095952)))
       (* x (+ 4.16438922228 (/ -110.1139242984811 x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.009) {
		tmp = x * 4.16438922228;
	} else if (x <= 6.2e-85) {
		tmp = (z * (x + -2.0)) / 47.066876606;
	} else if (x <= 2600000.0) {
		tmp = x * fma(x, fma(y, 0.3041881842569256, -5.843575199059173), (y * -0.0424927283095952));
	} else {
		tmp = x * (4.16438922228 + (-110.1139242984811 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.009)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= 6.2e-85)
		tmp = Float64(Float64(z * Float64(x + -2.0)) / 47.066876606);
	elseif (x <= 2600000.0)
		tmp = Float64(x * fma(x, fma(y, 0.3041881842569256, -5.843575199059173), Float64(y * -0.0424927283095952)));
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.009], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, 6.2e-85], N[(N[(z * N[(x + -2.0), $MachinePrecision]), $MachinePrecision] / 47.066876606), $MachinePrecision], If[LessEqual[x, 2600000.0], N[(x * N[(x * N[(y * 0.3041881842569256 + -5.843575199059173), $MachinePrecision] + N[(y * -0.0424927283095952), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.00899999999999999932

   1. Initial program 16.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 92.0

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

   5. Simplified 92.0%

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

   if -0.00899999999999999932 < x < 6.2000000000000005e-85

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 75.5

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

   5. Simplified 75.5%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 75.5%

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

   if 6.2000000000000005e-85 < x < 2.6e6

   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. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) + x \cdot \left(\frac{500000000}{23533438303} \cdot \left(y - \frac{4297481763}{15625000}\right) - \left(\frac{-131752537360500000000}{553822718361107519809} \cdot z + \frac{156699607947}{23533438303} \cdot \left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right)\right)\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)} \]

   5. Simplified 89.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0212463641547976, \mathsf{fma}\left(y, -0.0424927283095952, z \cdot 0.28294182010212804\right)\right), -6.658593866711955, \mathsf{fma}\left(z, 0.23789659216289816, \mathsf{fma}\left(y, 0.0212463641547976, -5.843575199059173\right)\right)\right), z \cdot 0.28294182010212804\right)\right), z \cdot -0.0424927283095952\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y + x \cdot \left(\left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) - \frac{137519416416}{23533438303}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1000000000}{23533438303} \cdot y + x \cdot \left(\left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) - \frac{137519416416}{23533438303}\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(\left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) - \frac{137519416416}{23533438303}\right) + \frac{-1000000000}{23533438303} \cdot y\right)} \]

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

         \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, \left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) - \frac{137519416416}{23533438303}, \frac{-1000000000}{23533438303} \cdot y\right)} \]

      4. sub-neg N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(\frac{500000000}{23533438303} \cdot y + \frac{156699607947000000000}{553822718361107519809} \cdot y\right) + \left(\mathsf{neg}\left(\frac{137519416416}{23533438303}\right)\right)}, \frac{-1000000000}{23533438303} \cdot y\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{y \cdot \left(\frac{500000000}{23533438303} + \frac{156699607947000000000}{553822718361107519809}\right)} + \left(\mathsf{neg}\left(\frac{137519416416}{23533438303}\right)\right), \frac{-1000000000}{23533438303} \cdot y\right) \]

      6. metadata-eval N/A

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

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

         \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(y, \frac{168466327098500000000}{553822718361107519809}, \mathsf{neg}\left(\frac{137519416416}{23533438303}\right)\right)}, \frac{-1000000000}{23533438303} \cdot y\right) \]

      8. metadata-eval N/A

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

      9. *-lowering-*.f64 60.0

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

   8. Simplified 60.0%

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

   if 2.6e6 < x

   1. Initial program 18.0%

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

   3. Taylor expanded in 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 83.8

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

   5. Simplified 83.8%

      \[\leadsto \color{blue}{x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.009:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{z \cdot \left(x + -2\right)}{47.066876606}\\ \mathbf{elif}\;x \leq 2600000:\\ \;\;\;\;x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(y, 0.3041881842569256, -5.843575199059173\right), y \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 92.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x, 4.16438922228, \frac{y}{x \cdot x}\right)\\ \mathbf{if}\;x \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, z \cdot 0.3041881842569256\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma x 4.16438922228 (/ y (* x x)))))
   (if (<= x -5.5)
     t_0
     (if (<= x 2.0)
       (fma
        x
        (fma -0.0424927283095952 y (* z 0.3041881842569256))
        (* z -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(x, 4.16438922228, (y / (x * x)));
	double tmp;
	if (x <= -5.5) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = fma(x, fma(-0.0424927283095952, y, (z * 0.3041881842569256)), (z * -0.0424927283095952));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(x, 4.16438922228, Float64(y / Float64(x * x)))
	tmp = 0.0
	if (x <= -5.5)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = fma(x, fma(-0.0424927283095952, y, Float64(z * 0.3041881842569256)), Float64(z * -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * 4.16438922228 + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5], t$95$0, If[LessEqual[x, 2.0], N[(x * N[(-0.0424927283095952 * y + N[(z * 0.3041881842569256), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5 or 2 < x

   1. Initial program 17.7%

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

   3. Taylor expanded in x around -inf

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

   5. Simplified 97.8%

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

   6. Taylor expanded in y around inf

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

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

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 97.8

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

   8. Simplified 97.8%

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

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}} - \frac{-104109730557}{25000000000} \cdot x} \]
   10. Step-by-step derivation

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

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

       2. metadata-eval N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

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

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

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

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

       7. unpow2 N/A

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

       8. *-lowering-*.f64 97.8

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

   11. Simplified 97.8%

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

   if -5.5 < x < 2

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) + x \cdot \left(\frac{500000000}{23533438303} \cdot \left(y - \frac{4297481763}{15625000}\right) - \left(\frac{-131752537360500000000}{553822718361107519809} \cdot z + \frac{156699607947}{23533438303} \cdot \left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right)\right)\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)} \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0212463641547976, \mathsf{fma}\left(y, -0.0424927283095952, z \cdot 0.28294182010212804\right)\right), -6.658593866711955, \mathsf{fma}\left(z, 0.23789659216289816, \mathsf{fma}\left(y, 0.0212463641547976, -5.843575199059173\right)\right)\right), z \cdot 0.28294182010212804\right)\right), z \cdot -0.0424927283095952\right)} \]

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. associate-+r+ N/A

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

      6. metadata-eval N/A

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

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

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

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

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

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

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

      10. metadata-eval N/A

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

      11. *-lowering-*.f64 94.5

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

   8. Simplified 94.5%

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

Alternative 15: 76.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -190:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-5}:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.787568985856513, 0.3041881842569256\right), -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -190.0)
   (* x 4.16438922228)
   (if (<= x 3.4e-5)
     (*
      z
      (fma
       x
       (fma x -1.787568985856513 0.3041881842569256)
       -0.0424927283095952))
     (* x (+ 4.16438922228 (/ -110.1139242984811 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -190.0) {
		tmp = x * 4.16438922228;
	} else if (x <= 3.4e-5) {
		tmp = z * fma(x, fma(x, -1.787568985856513, 0.3041881842569256), -0.0424927283095952);
	} else {
		tmp = x * (4.16438922228 + (-110.1139242984811 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -190.0)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= 3.4e-5)
		tmp = Float64(z * fma(x, fma(x, -1.787568985856513, 0.3041881842569256), -0.0424927283095952));
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -190.0], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, 3.4e-5], N[(z * N[(x * N[(x * -1.787568985856513 + 0.3041881842569256), $MachinePrecision] + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -190

   1. Initial program 16.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 92.0

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

   5. Simplified 92.0%

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

   if -190 < x < 3.4e-5

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) + x \cdot \left(\frac{500000000}{23533438303} \cdot \left(y - \frac{4297481763}{15625000}\right) - \left(\frac{-131752537360500000000}{553822718361107519809} \cdot z + \frac{156699607947}{23533438303} \cdot \left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right)\right)\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)} \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0212463641547976, \mathsf{fma}\left(y, -0.0424927283095952, z \cdot 0.28294182010212804\right)\right), -6.658593866711955, \mathsf{fma}\left(z, 0.23789659216289816, \mathsf{fma}\left(y, 0.0212463641547976, -5.843575199059173\right)\right)\right), z \cdot 0.28294182010212804\right)\right), z \cdot -0.0424927283095952\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x\right) - \frac{1000000000}{23533438303}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{z \cdot \left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x\right) - \frac{1000000000}{23533438303}\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1000000000}{23533438303}\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x\right) + \color{blue}{\frac{-1000000000}{23533438303}}\right) \]

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

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x, \frac{-1000000000}{23533438303}\right)} \]

      5. +-commutative N/A

         \[\leadsto z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x + \frac{168466327098500000000}{553822718361107519809}}, \frac{-1000000000}{23533438303}\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-23298017199368982832548000000000}{13033352773350869092174451844127}} + \frac{168466327098500000000}{553822718361107519809}, \frac{-1000000000}{23533438303}\right) \]

      7. accelerator-lowering-fma.f64 71.2

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

   8. Simplified 71.2%

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

   if 3.4e-5 < x

   1. Initial program 20.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} \]
   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 81.2

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

   5. Simplified 81.2%

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

Alternative 16: 76.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.12:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;z \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.787568985856513, 0.3041881842569256\right), -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.12)
   (* x 4.16438922228)
   (if (<= x 2.0)
     (*
      z
      (fma
       x
       (fma x -1.787568985856513 0.3041881842569256)
       -0.0424927283095952))
     (* x 4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.12) {
		tmp = x * 4.16438922228;
	} else if (x <= 2.0) {
		tmp = z * fma(x, fma(x, -1.787568985856513, 0.3041881842569256), -0.0424927283095952);
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.12)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= 2.0)
		tmp = Float64(z * fma(x, fma(x, -1.787568985856513, 0.3041881842569256), -0.0424927283095952));
	else
		tmp = Float64(x * 4.16438922228);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.12], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, 2.0], N[(z * N[(x * N[(x * -1.787568985856513 + 0.3041881842569256), $MachinePrecision] + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.12 or 2 < x

   1. Initial program 17.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 87.6

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

   5. Simplified 87.6%

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

   if -0.12 < x < 2

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{-1000000000}{23533438303} \cdot z + x \cdot \left(\left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) + x \cdot \left(\frac{500000000}{23533438303} \cdot \left(y - \frac{4297481763}{15625000}\right) - \left(\frac{-131752537360500000000}{553822718361107519809} \cdot z + \frac{156699607947}{23533438303} \cdot \left(\frac{500000000}{23533438303} \cdot \left(z + -2 \cdot y\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right)\right)\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)} \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.0212463641547976, \mathsf{fma}\left(y, -2, z\right), \mathsf{fma}\left(x, \mathsf{fma}\left(\mathsf{fma}\left(z, 0.0212463641547976, \mathsf{fma}\left(y, -0.0424927283095952, z \cdot 0.28294182010212804\right)\right), -6.658593866711955, \mathsf{fma}\left(z, 0.23789659216289816, \mathsf{fma}\left(y, 0.0212463641547976, -5.843575199059173\right)\right)\right), z \cdot 0.28294182010212804\right)\right), z \cdot -0.0424927283095952\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x\right) - \frac{1000000000}{23533438303}\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto \color{blue}{z \cdot \left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x\right) - \frac{1000000000}{23533438303}\right)} \]

      2. sub-neg N/A

         \[\leadsto z \cdot \color{blue}{\left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1000000000}{23533438303}\right)\right)\right)} \]

      3. metadata-eval N/A

         \[\leadsto z \cdot \left(x \cdot \left(\frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x\right) + \color{blue}{\frac{-1000000000}{23533438303}}\right) \]

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

         \[\leadsto z \cdot \color{blue}{\mathsf{fma}\left(x, \frac{168466327098500000000}{553822718361107519809} + \frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x, \frac{-1000000000}{23533438303}\right)} \]

      5. +-commutative N/A

         \[\leadsto z \cdot \mathsf{fma}\left(x, \color{blue}{\frac{-23298017199368982832548000000000}{13033352773350869092174451844127} \cdot x + \frac{168466327098500000000}{553822718361107519809}}, \frac{-1000000000}{23533438303}\right) \]

      6. *-commutative N/A

         \[\leadsto z \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-23298017199368982832548000000000}{13033352773350869092174451844127}} + \frac{168466327098500000000}{553822718361107519809}, \frac{-1000000000}{23533438303}\right) \]

      7. accelerator-lowering-fma.f64 70.7

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

   8. Simplified 70.7%

      \[\leadsto \color{blue}{z \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, -1.787568985856513, 0.3041881842569256\right), -0.0424927283095952\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 76.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 255:\\ \;\;\;\;\mathsf{fma}\left(-0.0424927283095952, z, x \cdot \left(z \cdot 0.3041881842569256\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.5)
   (* x 4.16438922228)
   (if (<= x 255.0)
     (fma -0.0424927283095952 z (* x (* z 0.3041881842569256)))
     (* 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 <= 255.0) {
		tmp = fma(-0.0424927283095952, z, (x * (z * 0.3041881842569256)));
	} 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 <= 255.0)
		tmp = fma(-0.0424927283095952, z, Float64(x * Float64(z * 0.3041881842569256)));
	else
		tmp = Float64(x * 4.16438922228);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.5], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, 255.0], N[(-0.0424927283095952 * z + N[(x * N[(z * 0.3041881842569256), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5 or 255 < x

   1. Initial program 17.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 87.6

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

   5. Simplified 87.6%

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

   if -5.5 < x < 255

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. +-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 70.9

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

   5. Simplified 70.9%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. metadata-eval N/A

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

      5. *-lowering-*.f64 70.5

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

   8. Simplified 70.5%

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

Alternative 18: 76.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.017:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.017)
   (* x 4.16438922228)
   (if (<= x 2.0) (* z -0.0424927283095952) (* x 4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.017) {
		tmp = x * 4.16438922228;
	} else if (x <= 2.0) {
		tmp = z * -0.0424927283095952;
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-0.017d0)) then
        tmp = x * 4.16438922228d0
    else if (x <= 2.0d0) then
        tmp = z * (-0.0424927283095952d0)
    else
        tmp = x * 4.16438922228d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.017) {
		tmp = x * 4.16438922228;
	} else if (x <= 2.0) {
		tmp = z * -0.0424927283095952;
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -0.017:
		tmp = x * 4.16438922228
	elif x <= 2.0:
		tmp = z * -0.0424927283095952
	else:
		tmp = x * 4.16438922228
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.017)
		tmp = Float64(x * 4.16438922228);
	elseif (x <= 2.0)
		tmp = Float64(z * -0.0424927283095952);
	else
		tmp = Float64(x * 4.16438922228);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -0.017)
		tmp = x * 4.16438922228;
	elseif (x <= 2.0)
		tmp = z * -0.0424927283095952;
	else
		tmp = x * 4.16438922228;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.017], N[(x * 4.16438922228), $MachinePrecision], If[LessEqual[x, 2.0], N[(z * -0.0424927283095952), $MachinePrecision], N[(x * 4.16438922228), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.017000000000000001 or 2 < x

   1. Initial program 17.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 87.6

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

   5. Simplified 87.6%

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

   if -0.017000000000000001 < x < 2

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 70.2

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

   5. Simplified 70.2%

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

Alternative 19: 44.4% 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 59.6%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 44.6

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

5. Simplified 44.6%

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

Developer Target 1: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024198 
(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)))