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

Percentage Accurate: 59.2% → 98.6%

Time: 16.4s

Alternatives: 21

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: 59.2% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), x \cdot x, \mathsf{fma}\left(x, 313.399215894, 47.066876606\right)\right)}}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, -4.16438922228, x \cdot \frac{110.1139242984811 + \frac{\frac{130977.50649958357 - y}{x} + -3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          x
          (+
           (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
           y))
         z))
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606))
      1e+306)
   (/
    (*
     (fma x x -4.0)
     (/
      (fma
       x
       (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
       z)
      (fma
       (fma x (+ x 43.3400022514) 263.505074721)
       (* x x)
       (fma x 313.399215894 47.066876606))))
    (+ x 2.0))
   (-
    (fma
     x
     -4.16438922228
     (*
      x
      (/
       (+
        110.1139242984811
        (/ (+ (/ (- 130977.50649958357 y) x) -3655.1204654076414) x))
       x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+306) {
		tmp = (fma(x, x, -4.0) * (fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(fma(x, (x + 43.3400022514), 263.505074721), (x * x), fma(x, 313.399215894, 47.066876606)))) / (x + 2.0);
	} else {
		tmp = -fma(x, -4.16438922228, (x * ((110.1139242984811 + ((((130977.50649958357 - y) / x) + -3655.1204654076414) / x)) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+306)
		tmp = Float64(Float64(fma(x, x, -4.0) * Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(fma(x, Float64(x + 43.3400022514), 263.505074721), Float64(x * x), fma(x, 313.399215894, 47.066876606)))) / Float64(x + 2.0));
	else
		tmp = Float64(-fma(x, -4.16438922228, Float64(x * Float64(Float64(110.1139242984811 + Float64(Float64(Float64(Float64(130977.50649958357 - y) / x) + -3655.1204654076414) / x)) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], 1e+306], N[(N[(N[(x * x + -4.0), $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] / N[(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[(x + 2.0), $MachinePrecision]), $MachinePrecision], (-N[(x * -4.16438922228 + N[(x * N[(N[(110.1139242984811 + N[(N[(N[(N[(130977.50649958357 - y), $MachinePrecision] / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.4%

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

   4. Applied rewrites 98.3%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. distribute-rgt-in N/A

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

      6. associate-+l+ N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 98.3

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

   6. Applied rewrites 98.3%

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

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

   1. Initial program 0.6%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), x \cdot x, \mathsf{fma}\left(x, 313.399215894, 47.066876606\right)\right)}}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, -4.16438922228, x \cdot \frac{110.1139242984811 + \frac{\frac{130977.50649958357 - y}{x} + -3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, -4.16438922228, x \cdot \frac{110.1139242984811 + \frac{\frac{130977.50649958357 - y}{x} + -3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          x
          (+
           (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
           y))
         z))
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606))
      1e+306)
   (/
    (*
     (fma x x -4.0)
     (/
      (fma
       x
       (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
       z)
      (fma
       x
       (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
       47.066876606)))
    (+ x 2.0))
   (-
    (fma
     x
     -4.16438922228
     (*
      x
      (/
       (+
        110.1139242984811
        (/ (+ (/ (- 130977.50649958357 y) x) -3655.1204654076414) x))
       x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+306) {
		tmp = (fma(x, x, -4.0) * (fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606))) / (x + 2.0);
	} else {
		tmp = -fma(x, -4.16438922228, (x * ((110.1139242984811 + ((((130977.50649958357 - y) / x) + -3655.1204654076414) / x)) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+306)
		tmp = Float64(Float64(fma(x, x, -4.0) * Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606))) / Float64(x + 2.0));
	else
		tmp = Float64(-fma(x, -4.16438922228, Float64(x * Float64(Float64(110.1139242984811 + Float64(Float64(Float64(Float64(130977.50649958357 - y) / x) + -3655.1204654076414) / x)) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], 1e+306], N[(N[(N[(x * x + -4.0), $MachinePrecision] * N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 2.0), $MachinePrecision]), $MachinePrecision], (-N[(x * -4.16438922228 + N[(x * N[(N[(110.1139242984811 + N[(N[(N[(N[(130977.50649958357 - y), $MachinePrecision] / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.4%

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

   4. Applied rewrites 98.3%

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

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

   1. Initial program 0.6%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{x + 2}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, -4.16438922228, x \cdot \frac{110.1139242984811 + \frac{\frac{130977.50649958357 - y}{x} + -3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, -4.16438922228, x \cdot \frac{110.1139242984811 + \frac{\frac{130977.50649958357 - y}{x} + -3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          x
          (+
           (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
           y))
         z))
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606))
      1e+306)
   (/
    (fma
     x
     (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
     z)
    (/
     (fma
      x
      (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
      47.066876606)
     (+ x -2.0)))
   (-
    (fma
     x
     -4.16438922228
     (*
      x
      (/
       (+
        110.1139242984811
        (/ (+ (/ (- 130977.50649958357 y) x) -3655.1204654076414) x))
       x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+306) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / (fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606) / (x + -2.0));
	} else {
		tmp = -fma(x, -4.16438922228, (x * ((110.1139242984811 + ((((130977.50649958357 - y) / x) + -3655.1204654076414) / x)) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+306)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / Float64(fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606) / Float64(x + -2.0)));
	else
		tmp = Float64(-fma(x, -4.16438922228, Float64(x * Float64(Float64(110.1139242984811 + Float64(Float64(Float64(Float64(130977.50649958357 - y) / x) + -3655.1204654076414) / x)) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], 1e+306], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] / N[(N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision] / N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(x * -4.16438922228 + N[(x * N[(N[(110.1139242984811 + N[(N[(N[(N[(130977.50649958357 - y), $MachinePrecision] / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.4%

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

   4. Applied rewrites 98.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-fma.f64 N/A

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

   6. Applied rewrites 98.3%

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

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

   1. Initial program 0.6%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, -4.16438922228, x \cdot \frac{110.1139242984811 + \frac{\frac{130977.50649958357 - y}{x} + -3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)} \cdot \left(x + -2\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, -4.16438922228, x \cdot \frac{110.1139242984811 + \frac{\frac{130977.50649958357 - y}{x} + -3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          x
          (+
           (* x (+ (* x (+ (* x 4.16438922228) 78.6994924154)) 137.519416416))
           y))
         z))
       (+
        (*
         x
         (+ (* x (+ (* x (+ x 43.3400022514)) 263.505074721)) 313.399215894))
        47.066876606))
      1e+306)
   (*
    (/
     (fma
      x
      (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
      z)
     (fma
      x
      (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
      47.066876606))
    (+ x -2.0))
   (-
    (fma
     x
     -4.16438922228
     (*
      x
      (/
       (+
        110.1139242984811
        (/ (+ (/ (- 130977.50649958357 y) x) -3655.1204654076414) x))
       x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((x * ((x * ((x * ((x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / ((x * ((x * ((x * (x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+306) {
		tmp = (fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)) * (x + -2.0);
	} else {
		tmp = -fma(x, -4.16438922228, (x * ((110.1139242984811 + ((((130977.50649958357 - y) / x) + -3655.1204654076414) / x)) / x)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * 4.16438922228) + 78.6994924154)) + 137.519416416)) + y)) + z)) / Float64(Float64(x * Float64(Float64(x * Float64(Float64(x * Float64(x + 43.3400022514)) + 263.505074721)) + 313.399215894)) + 47.066876606)) <= 1e+306)
		tmp = Float64(Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)) * Float64(x + -2.0));
	else
		tmp = Float64(-fma(x, -4.16438922228, Float64(x * Float64(Float64(110.1139242984811 + Float64(Float64(Float64(Float64(130977.50649958357 - y) / x) + -3655.1204654076414) / x)) / x))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(x * N[(N[(x * N[(N[(x * N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision]), $MachinePrecision] + 137.519416416), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(N[(x * N[(N[(x * N[(x + 43.3400022514), $MachinePrecision]), $MachinePrecision] + 263.505074721), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision]), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], 1e+306], N[(N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision] * N[(x + -2.0), $MachinePrecision]), $MachinePrecision], (-N[(x * -4.16438922228 + N[(x * N[(N[(110.1139242984811 + N[(N[(N[(N[(130977.50649958357 - y), $MachinePrecision] / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.4%

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

   4. Applied rewrites 98.3%

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

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

   1. Initial program 0.6%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 97.5%

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

   7. Applied rewrites 97.5%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+306}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)} \cdot \left(x + -2\right)\\ \mathbf{else}:\\ \;\;\;\;-\mathsf{fma}\left(x, -4.16438922228, x \cdot \frac{110.1139242984811 + \frac{\frac{130977.50649958357 - y}{x} + -3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -37000:\\ \;\;\;\;-\mathsf{fma}\left(x, -4.16438922228, x \cdot \frac{110.1139242984811 + \frac{\frac{130977.50649958357 - y}{x} + -3655.1204654076414}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 78.6994924154, 137.519416416\right), y\right), z\right) \cdot \frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y}{x}}{x}}{x} - -4.16438922228\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -37000.0)
   (-
    (fma
     x
     -4.16438922228
     (*
      x
      (/
       (+
        110.1139242984811
        (/ (+ (/ (- 130977.50649958357 y) x) -3655.1204654076414) x))
       x))))
   (if (<= x 1.55e+30)
     (*
      (fma x (fma x (fma x 78.6994924154 137.519416416) y) z)
      (/
       (+ x -2.0)
       (fma
        x
        (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
        47.066876606)))
     (* x (- (/ (/ (/ y x) x) x) -4.16438922228)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -37000.0) {
		tmp = -fma(x, -4.16438922228, (x * ((110.1139242984811 + ((((130977.50649958357 - y) / x) + -3655.1204654076414) / x)) / x)));
	} else if (x <= 1.55e+30) {
		tmp = fma(x, fma(x, fma(x, 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 = x * ((((y / x) / x) / x) - -4.16438922228);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -37000.0)
		tmp = Float64(-fma(x, -4.16438922228, Float64(x * Float64(Float64(110.1139242984811 + Float64(Float64(Float64(Float64(130977.50649958357 - y) / x) + -3655.1204654076414) / x)) / x))));
	elseif (x <= 1.55e+30)
		tmp = Float64(fma(x, fma(x, fma(x, 78.6994924154, 137.519416416), y), z) * Float64(Float64(x + -2.0) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)));
	else
		tmp = Float64(x * Float64(Float64(Float64(Float64(y / x) / x) / x) - -4.16438922228));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -37000.0], (-N[(x * -4.16438922228 + N[(x * N[(N[(110.1139242984811 + N[(N[(N[(N[(130977.50649958357 - y), $MachinePrecision] / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[x, 1.55e+30], N[(N[(x * N[(x * N[(x * 78.6994924154 + 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 * N[(N[(N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -37000

   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. Applied rewrites 96.1%

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

   7. Applied rewrites 96.1%

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

   if -37000 < x < 1.5499999999999999e30

   1. Initial program 98.3%

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 97.5

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

   7. Applied rewrites 97.5%

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

   if 1.5499999999999999e30 < x

   1. Initial program 8.3%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.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. lower-*.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. lower-*.f64 97.8

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

   8. Applied rewrites 97.8%

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

      1. lift-*.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.9

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

   10. Applied rewrites 97.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -37000:\\ \;\;\;\;-\mathsf{fma}\left(x, -4.16438922228, x \cdot \frac{110.1139242984811 + \frac{\frac{130977.50649958357 - y}{x} + -3655.1204654076414}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 78.6994924154, 137.519416416\right), y\right), z\right) \cdot \frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y}{x}}{x}}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -37000:\\ \;\;\;\;-\mathsf{fma}\left(x, -4.16438922228, x \cdot \frac{110.1139242984811 + \frac{\frac{130977.50649958357 - y}{x} + -3655.1204654076414}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+30}:\\ \;\;\;\;\frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)} \cdot \mathsf{fma}\left(x, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y}{x}}{x}}{x} - -4.16438922228\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -37000.0)
   (-
    (fma
     x
     -4.16438922228
     (*
      x
      (/
       (+
        110.1139242984811
        (/ (+ (/ (- 130977.50649958357 y) x) -3655.1204654076414) x))
       x))))
   (if (<= x 1.55e+30)
     (*
      (/
       (+ x -2.0)
       (fma
        x
        (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
        47.066876606))
      (fma x y z))
     (* x (- (/ (/ (/ y x) x) x) -4.16438922228)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -37000.0) {
		tmp = -fma(x, -4.16438922228, (x * ((110.1139242984811 + ((((130977.50649958357 - y) / x) + -3655.1204654076414) / x)) / x)));
	} else if (x <= 1.55e+30) {
		tmp = ((x + -2.0) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)) * fma(x, y, z);
	} else {
		tmp = x * ((((y / x) / x) / x) - -4.16438922228);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -37000.0)
		tmp = Float64(-fma(x, -4.16438922228, Float64(x * Float64(Float64(110.1139242984811 + Float64(Float64(Float64(Float64(130977.50649958357 - y) / x) + -3655.1204654076414) / x)) / x))));
	elseif (x <= 1.55e+30)
		tmp = Float64(Float64(Float64(x + -2.0) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)) * fma(x, y, z));
	else
		tmp = Float64(x * Float64(Float64(Float64(Float64(y / x) / x) / x) - -4.16438922228));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -37000.0], (-N[(x * -4.16438922228 + N[(x * N[(N[(110.1139242984811 + N[(N[(N[(N[(130977.50649958357 - y), $MachinePrecision] / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[x, 1.55e+30], N[(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] * N[(x * y + z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -37000

   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. Applied rewrites 96.1%

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

   7. Applied rewrites 96.1%

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

   if -37000 < x < 1.5499999999999999e30

   1. Initial program 98.3%

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.5

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

   7. Applied rewrites 96.5%

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

   if 1.5499999999999999e30 < x

   1. Initial program 8.3%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.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. lower-*.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. lower-*.f64 97.8

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

   8. Applied rewrites 97.8%

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

      1. lift-*.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.9

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

   10. Applied rewrites 97.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -37000:\\ \;\;\;\;-\mathsf{fma}\left(x, -4.16438922228, x \cdot \frac{110.1139242984811 + \frac{\frac{130977.50649958357 - y}{x} + -3655.1204654076414}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+30}:\\ \;\;\;\;\frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)} \cdot \mathsf{fma}\left(x, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y}{x}}{x}}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -37000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{\frac{y - 130977.50649958357}{x} - -3655.1204654076414}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+30}:\\ \;\;\;\;\frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)} \cdot \mathsf{fma}\left(x, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y}{x}}{x}}{x} - -4.16438922228\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -37000.0)
   (*
    x
    (+
     4.16438922228
     (/
      (+
       -110.1139242984811
       (/ (- (/ (- y 130977.50649958357) x) -3655.1204654076414) x))
      x)))
   (if (<= x 1.55e+30)
     (*
      (/
       (+ x -2.0)
       (fma
        x
        (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
        47.066876606))
      (fma x y z))
     (* x (- (/ (/ (/ y x) x) x) -4.16438922228)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -37000.0) {
		tmp = x * (4.16438922228 + ((-110.1139242984811 + ((((y - 130977.50649958357) / x) - -3655.1204654076414) / x)) / x));
	} else if (x <= 1.55e+30) {
		tmp = ((x + -2.0) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)) * fma(x, y, z);
	} else {
		tmp = x * ((((y / x) / x) / x) - -4.16438922228);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -37000.0)
		tmp = Float64(x * Float64(4.16438922228 + Float64(Float64(-110.1139242984811 + Float64(Float64(Float64(Float64(y - 130977.50649958357) / x) - -3655.1204654076414) / x)) / x)));
	elseif (x <= 1.55e+30)
		tmp = Float64(Float64(Float64(x + -2.0) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)) * fma(x, y, z));
	else
		tmp = Float64(x * Float64(Float64(Float64(Float64(y / x) / x) / x) - -4.16438922228));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -37000.0], N[(x * N[(4.16438922228 + N[(N[(-110.1139242984811 + N[(N[(N[(N[(y - 130977.50649958357), $MachinePrecision] / x), $MachinePrecision] - -3655.1204654076414), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e+30], N[(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] * N[(x * y + z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -37000

   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. Applied rewrites 96.1%

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

   7. Applied rewrites 96.1%

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

   if -37000 < x < 1.5499999999999999e30

   1. Initial program 98.3%

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.5

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

   7. Applied rewrites 96.5%

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

   if 1.5499999999999999e30 < x

   1. Initial program 8.3%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.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. lower-*.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. lower-*.f64 97.8

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

   8. Applied rewrites 97.8%

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

      1. lift-*.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.9

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

   10. Applied rewrites 97.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -37000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{\frac{y - 130977.50649958357}{x} - -3655.1204654076414}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+30}:\\ \;\;\;\;\frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)} \cdot \mathsf{fma}\left(x, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y}{x}}{x}}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\frac{\frac{\frac{y}{x}}{x}}{x} - -4.16438922228\right)\\ \mathbf{if}\;x \leq -37000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+30}:\\ \;\;\;\;\frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)} \cdot \mathsf{fma}\left(x, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- (/ (/ (/ y x) x) x) -4.16438922228))))
   (if (<= x -37000.0)
     t_0
     (if (<= x 1.55e+30)
       (*
        (/
         (+ x -2.0)
         (fma
          x
          (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
          47.066876606))
        (fma x y z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * ((((y / x) / x) / x) - -4.16438922228);
	double tmp;
	if (x <= -37000.0) {
		tmp = t_0;
	} else if (x <= 1.55e+30) {
		tmp = ((x + -2.0) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)) * fma(x, y, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(Float64(Float64(Float64(y / x) / x) / x) - -4.16438922228))
	tmp = 0.0
	if (x <= -37000.0)
		tmp = t_0;
	elseif (x <= 1.55e+30)
		tmp = Float64(Float64(Float64(x + -2.0) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)) * fma(x, y, z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[(N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -37000.0], t$95$0, If[LessEqual[x, 1.55e+30], N[(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] * N[(x * y + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -37000 or 1.5499999999999999e30 < x

   1. Initial program 12.6%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 97.0%

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

   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. lower-/.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. lower-*.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. lower-*.f64 96.5

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

   8. Applied rewrites 96.5%

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

      1. lift-*.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 96.5

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

   10. Applied rewrites 96.5%

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

   if -37000 < x < 1.5499999999999999e30

   1. Initial program 98.3%

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

   4. Applied rewrites 98.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.5

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

   7. Applied rewrites 96.5%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -37000:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y}{x}}{x}}{x} - -4.16438922228\right)\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+30}:\\ \;\;\;\;\frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)} \cdot \mathsf{fma}\left(x, y, z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y}{x}}{x}}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(\frac{\frac{\frac{y}{x}}{x}}{x} - -4.16438922228\right)\\ \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 32:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (- (/ (/ (/ y x) x) x) -4.16438922228))))
   (if (<= x -0.175)
     t_0
     (if (<= x 32.0)
       (*
        (fma
         x
         (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
         z)
        (fma x 0.3041881842569256 -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * ((((y / x) / x) / x) - -4.16438922228);
	double tmp;
	if (x <= -0.175) {
		tmp = t_0;
	} else if (x <= 32.0) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, 0.3041881842569256, -0.0424927283095952);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(Float64(Float64(Float64(y / x) / x) / x) - -4.16438922228))
	tmp = 0.0
	if (x <= -0.175)
		tmp = t_0;
	elseif (x <= 32.0)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, 0.3041881842569256, -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(N[(N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.175], t$95$0, If[LessEqual[x, 32.0], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * N[(x * 0.3041881842569256 + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.17499999999999999 or 32 < x

   1. Initial program 17.1%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 94.2%

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

   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. lower-/.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. lower-*.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. lower-*.f64 93.4

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

   8. Applied rewrites 93.4%

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

      1. lift-*.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 93.5

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

   10. Applied rewrites 93.5%

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

   if -0.17499999999999999 < x < 32

   1. Initial program 99.0%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \color{blue}{\left(\frac{168466327098500000000}{553822718361107519809} \cdot x - \frac{1000000000}{23533438303}\right)} \]
   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. lower-fma.f64 96.2

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

   7. Applied rewrites 96.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y}{x}}{x}}{x} - -4.16438922228\right)\\ \mathbf{elif}\;x \leq 32:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{\frac{\frac{y}{x}}{x}}{x} - -4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;x \cdot 4.16438922228 + \frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq 32:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.175)
   (+ (* x 4.16438922228) (/ y (* x x)))
   (if (<= x 32.0)
     (*
      (fma
       x
       (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
       z)
      (fma x 0.3041881842569256 -0.0424927283095952))
     (* x (+ 4.16438922228 (/ y (* x (* x x))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.175) {
		tmp = (x * 4.16438922228) + (y / (x * x));
	} else if (x <= 32.0) {
		tmp = fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, 0.3041881842569256, -0.0424927283095952);
	} else {
		tmp = x * (4.16438922228 + (y / (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.175)
		tmp = Float64(Float64(x * 4.16438922228) + Float64(y / Float64(x * x)));
	elseif (x <= 32.0)
		tmp = Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) * fma(x, 0.3041881842569256, -0.0424927283095952));
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(y / Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.175], N[(N[(x * 4.16438922228), $MachinePrecision] + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 32.0], N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] * N[(x * 0.3041881842569256 + -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * N[(4.16438922228 + N[(y / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.17499999999999999

   1. Initial program 19.7%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   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. Applied rewrites 93.0%

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

   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. lower-/.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. lower-*.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. lower-*.f64 92.1

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

   8. Applied rewrites 92.1%

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

   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. lower-+.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 92.1

          \[\leadsto \frac{y}{x \cdot x} + \color{blue}{x \cdot 4.16438922228} \]

   11. Applied rewrites 92.1%

       \[\leadsto \color{blue}{\frac{y}{x \cdot x} + x \cdot 4.16438922228} \]

   if -0.17499999999999999 < x < 32

   1. Initial program 99.0%

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{104109730557}{25000000000}, \frac{393497462077}{5000000000}\right), \frac{4297481763}{31250000}\right), y\right), z\right) \cdot \color{blue}{\left(\frac{168466327098500000000}{553822718361107519809} \cdot x - \frac{1000000000}{23533438303}\right)} \]
   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. lower-fma.f64 96.2

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

   7. Applied rewrites 96.2%

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

   if 32 < x

   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. Applied rewrites 95.2%

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

   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. lower-/.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. lower-*.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. lower-*.f64 94.7

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

   8. Applied rewrites 94.7%

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

      1. lift-*.f64 N/A

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

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

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

      4. lift--.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. distribute-frac-neg N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f64 N/A

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

      17. lower-+.f64 94.7

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

   10. Applied rewrites 94.7%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;x \cdot 4.16438922228 + \frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq 32:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right) \cdot \mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 94.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;x \cdot 4.16438922228 + \frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, x \cdot -5.843575199059173\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.175)
   (+ (* x 4.16438922228) (/ y (* x x)))
   (if (<= x 2.0)
     (fma
      x
      (fma -0.0424927283095952 y (* x -5.843575199059173))
      (* z -0.0424927283095952))
     (* x (+ 4.16438922228 (/ y (* x (* x x))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.175) {
		tmp = (x * 4.16438922228) + (y / (x * x));
	} else if (x <= 2.0) {
		tmp = fma(x, fma(-0.0424927283095952, y, (x * -5.843575199059173)), (z * -0.0424927283095952));
	} else {
		tmp = x * (4.16438922228 + (y / (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.175)
		tmp = Float64(Float64(x * 4.16438922228) + Float64(y / Float64(x * x)));
	elseif (x <= 2.0)
		tmp = fma(x, fma(-0.0424927283095952, y, Float64(x * -5.843575199059173)), Float64(z * -0.0424927283095952));
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(y / Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.175], N[(N[(x * 4.16438922228), $MachinePrecision] + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.0], N[(x * N[(-0.0424927283095952 * y + N[(x * -5.843575199059173), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * N[(4.16438922228 + N[(y / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.17499999999999999

   1. Initial program 19.7%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   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. Applied rewrites 93.0%

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

   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. lower-/.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. lower-*.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. lower-*.f64 92.1

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

   8. Applied rewrites 92.1%

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

   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. lower-+.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 92.1

          \[\leadsto \frac{y}{x \cdot x} + \color{blue}{x \cdot 4.16438922228} \]

   11. Applied rewrites 92.1%

       \[\leadsto \color{blue}{\frac{y}{x \cdot x} + x \cdot 4.16438922228} \]

   if -0.17499999999999999 < x < 2

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 95.8

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

   11. Applied rewrites 95.8%

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

   if 2 < x

   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. Applied rewrites 95.2%

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

   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. lower-/.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. lower-*.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. lower-*.f64 94.7

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

   8. Applied rewrites 94.7%

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

      1. lift-*.f64 N/A

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

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

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

      4. lift--.f64 N/A

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

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

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. distribute-neg-frac2 N/A

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

      14. distribute-frac-neg N/A

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

      15. frac-2neg N/A

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

      16. lift-/.f64 N/A

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

      17. lower-+.f64 94.7

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

   10. Applied rewrites 94.7%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;x \cdot 4.16438922228 + \frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, x \cdot -5.843575199059173\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{y}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 94.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot 4.16438922228 + \frac{y}{x \cdot x}\\ \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, x \cdot -5.843575199059173\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* x 4.16438922228) (/ y (* x x)))))
   (if (<= x -0.175)
     t_0
     (if (<= x 2.0)
       (fma
        x
        (fma -0.0424927283095952 y (* x -5.843575199059173))
        (* z -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 4.16438922228) + (y / (x * x));
	double tmp;
	if (x <= -0.175) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = fma(x, fma(-0.0424927283095952, y, (x * -5.843575199059173)), (z * -0.0424927283095952));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 4.16438922228) + Float64(y / Float64(x * x)))
	tmp = 0.0
	if (x <= -0.175)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = fma(x, fma(-0.0424927283095952, y, Float64(x * -5.843575199059173)), Float64(z * -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 4.16438922228), $MachinePrecision] + N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.175], t$95$0, If[LessEqual[x, 2.0], N[(x * N[(-0.0424927283095952 * y + N[(x * -5.843575199059173), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.17499999999999999 or 2 < x

   1. Initial program 17.1%

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

   3. Taylor expanded in x around -inf

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

   5. Applied rewrites 94.2%

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

   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. lower-/.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. lower-*.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. lower-*.f64 93.4

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

   8. Applied rewrites 93.4%

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

   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. lower-+.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 93.4

          \[\leadsto \frac{y}{x \cdot x} + \color{blue}{x \cdot 4.16438922228} \]

   11. Applied rewrites 93.4%

       \[\leadsto \color{blue}{\frac{y}{x \cdot x} + x \cdot 4.16438922228} \]

   if -0.17499999999999999 < x < 2

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 95.8

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

   11. Applied rewrites 95.8%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;x \cdot 4.16438922228 + \frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, x \cdot -5.843575199059173\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228 + \frac{y}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 92.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, x \cdot -5.843575199059173\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.175)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 2.0)
     (fma
      x
      (fma -0.0424927283095952 y (* x -5.843575199059173))
      (* z -0.0424927283095952))
     (fma x 4.16438922228 -110.1139242984811))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.175) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 2.0) {
		tmp = fma(x, fma(-0.0424927283095952, y, (x * -5.843575199059173)), (z * -0.0424927283095952));
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.175)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 2.0)
		tmp = fma(x, fma(-0.0424927283095952, y, Float64(x * -5.843575199059173)), Float64(z * -0.0424927283095952));
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.175], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 2.0], N[(x * N[(-0.0424927283095952 * y + N[(x * -5.843575199059173), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.17499999999999999 or 2 < x

   1. Initial program 17.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 19.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 59.2

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

   8. Applied rewrites 59.2%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 85.8

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

   11. Applied rewrites 85.8%

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

   if -0.17499999999999999 < x < 2

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 97.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.1%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 95.8

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

   11. Applied rewrites 95.8%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.175:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, x \cdot -5.843575199059173\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0075:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-75}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 7.2:\\ \;\;\;\;x \cdot \left(y \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.0075)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 8.8e-75)
     (* z -0.0424927283095952)
     (if (<= x 7.2)
       (* x (* y -0.0424927283095952))
       (fma x 4.16438922228 -110.1139242984811)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0075) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 8.8e-75) {
		tmp = z * -0.0424927283095952;
	} else if (x <= 7.2) {
		tmp = x * (y * -0.0424927283095952);
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.0075)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 8.8e-75)
		tmp = Float64(z * -0.0424927283095952);
	elseif (x <= 7.2)
		tmp = Float64(x * Float64(y * -0.0424927283095952));
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.0075], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 8.8e-75], N[(z * -0.0424927283095952), $MachinePrecision], If[LessEqual[x, 7.2], N[(x * N[(y * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0074999999999999997 or 7.20000000000000018 < x

   1. Initial program 17.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 19.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 58.9

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

   8. Applied rewrites 58.9%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 85.3

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

   11. Applied rewrites 85.3%

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

   if -0.0074999999999999997 < x < 8.80000000000000022e-75

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.8

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

   5. Applied rewrites 69.8%

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

   if 8.80000000000000022e-75 < x < 7.20000000000000018

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 56.5

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

   11. Applied rewrites 56.5%

       \[\leadsto \color{blue}{-0.0424927283095952 \cdot \left(x \cdot y\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 56.6

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

   13. Applied rewrites 56.6%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0075:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-75}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 7.2:\\ \;\;\;\;x \cdot \left(y \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0075:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-75}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{elif}\;x \leq 7.2:\\ \;\;\;\;-0.0424927283095952 \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.0075)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 8.8e-75)
     (* z -0.0424927283095952)
     (if (<= x 7.2)
       (* -0.0424927283095952 (* x y))
       (fma x 4.16438922228 -110.1139242984811)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0075) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 8.8e-75) {
		tmp = z * -0.0424927283095952;
	} else if (x <= 7.2) {
		tmp = -0.0424927283095952 * (x * y);
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.0075)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 8.8e-75)
		tmp = Float64(z * -0.0424927283095952);
	elseif (x <= 7.2)
		tmp = Float64(-0.0424927283095952 * Float64(x * y));
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.0075], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 8.8e-75], N[(z * -0.0424927283095952), $MachinePrecision], If[LessEqual[x, 7.2], N[(-0.0424927283095952 * N[(x * y), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0074999999999999997 or 7.20000000000000018 < x

   1. Initial program 17.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 19.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 58.9

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

   8. Applied rewrites 58.9%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 85.3

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

   11. Applied rewrites 85.3%

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

   if -0.0074999999999999997 < x < 8.80000000000000022e-75

   1. Initial program 98.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.8

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

   5. Applied rewrites 69.8%

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

   if 8.80000000000000022e-75 < x < 7.20000000000000018

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 73.2%

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

   9. Taylor expanded in y around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 56.5

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

   11. Applied rewrites 56.5%

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

Alternative 16: 89.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.015:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 4.3:\\ \;\;\;\;\mathsf{fma}\left(x, y, z\right) \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.015)
   (* x (+ 4.16438922228 (/ -110.1139242984811 x)))
   (if (<= x 4.3)
     (* (fma x y z) -0.0424927283095952)
     (fma x 4.16438922228 -110.1139242984811))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.015) {
		tmp = x * (4.16438922228 + (-110.1139242984811 / x));
	} else if (x <= 4.3) {
		tmp = fma(x, y, z) * -0.0424927283095952;
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.015)
		tmp = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)));
	elseif (x <= 4.3)
		tmp = Float64(fma(x, y, z) * -0.0424927283095952);
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.015], N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.3], N[(N[(x * y + z), $MachinePrecision] * -0.0424927283095952), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.014999999999999999

   1. Initial program 20.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\frac{104109730557}{25000000000} - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} \]
   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. lower-*.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. lower-+.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. lower-/.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 82.2

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

   5. Applied rewrites 82.2%

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

   if -0.014999999999999999 < x < 4.29999999999999982

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.7%

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

   9. Taylor expanded in x around 0

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

       1. distribute-lft-out N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 95.7

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

   11. Applied rewrites 95.7%

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

   if 4.29999999999999982 < x

   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 z around inf

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

   5. Applied rewrites 17.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 60.1

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

   8. Applied rewrites 60.1%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 88.1

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

   11. Applied rewrites 88.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.015:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 4.3:\\ \;\;\;\;\mathsf{fma}\left(x, y, z\right) \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 89.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.015:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 4.3:\\ \;\;\;\;\mathsf{fma}\left(x, y, z\right) \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.015)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 4.3)
     (* (fma x y z) -0.0424927283095952)
     (fma x 4.16438922228 -110.1139242984811))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.015) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 4.3) {
		tmp = fma(x, y, z) * -0.0424927283095952;
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.015)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 4.3)
		tmp = Float64(fma(x, y, z) * -0.0424927283095952);
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.015], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 4.3], N[(N[(x * y + z), $MachinePrecision] * -0.0424927283095952), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.014999999999999999 or 4.29999999999999982 < x

   1. Initial program 17.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 19.9%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 58.9

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

   8. Applied rewrites 58.9%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 85.3

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

   11. Applied rewrites 85.3%

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

   if -0.014999999999999999 < x < 4.29999999999999982

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.7%

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

   9. Taylor expanded in x around 0

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

       1. distribute-lft-out N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 95.7

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

   11. Applied rewrites 95.7%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.015:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 4.3:\\ \;\;\;\;\mathsf{fma}\left(x, y, z\right) \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 76.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0075:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \mathbf{elif}\;x \leq 0.025:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922228, -110.1139242984811\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.0075)
   (fma x 4.16438922228 -110.1139242984811)
   (if (<= x 0.025)
     (* z -0.0424927283095952)
     (fma x 4.16438922228 -110.1139242984811))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -0.0075) {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	} else if (x <= 0.025) {
		tmp = z * -0.0424927283095952;
	} else {
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -0.0075)
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	elseif (x <= 0.025)
		tmp = Float64(z * -0.0424927283095952);
	else
		tmp = fma(x, 4.16438922228, -110.1139242984811);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -0.0075], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision], If[LessEqual[x, 0.025], N[(z * -0.0424927283095952), $MachinePrecision], N[(x * 4.16438922228 + -110.1139242984811), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0074999999999999997 or 0.025000000000000001 < x

   1. Initial program 19.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 20.4%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-+.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 58.1

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

   8. Applied rewrites 58.1%

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

   9. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 84.1

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

   11. Applied rewrites 84.1%

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

   if -0.0074999999999999997 < x < 0.025000000000000001

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.8

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

   5. Applied rewrites 65.8%

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

Alternative 19: 76.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0075:\\ \;\;\;\;x \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -0.0075)
   (* 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 <= -0.0075) {
		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 <= (-0.0075d0)) 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 <= -0.0075) {
		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 <= -0.0075:
		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 <= -0.0075)
		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 <= -0.0075)
		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, -0.0075], 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 < -0.0074999999999999997 or 2 < x

   1. Initial program 17.8%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if -0.0074999999999999997 < x < 2

   1. Initial program 99.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 64.7

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

   5. Applied rewrites 64.7%

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

Alternative 20: 44.6% 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 57.1%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 45.4

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

5. Applied rewrites 45.4%

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

Alternative 21: 3.4% accurate, 79.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in z around inf

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

5. Applied rewrites 57.2%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-+.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. distribute-neg-frac N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 31.9

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

8. Applied rewrites 31.9%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 3.3%

    \[\leadsto \color{blue}{-110.1139242984811} \]
12. Add Preprocessing

Developer Target 1: 98.7% 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 2024219 
(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)))