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

Percentage Accurate: 57.7% → 97.5%

Time: 17.8s

Alternatives: 20

Speedup: 4.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+282}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), x \cdot x, \mathsf{fma}\left(x, 313.399215894, 47.066876606\right)\right)}}{\frac{1}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{\frac{3451.550173699799 + \frac{y - 124074.40615218398}{x}}{x} - 101.7851458539211}{x}\right)\\ \end{array} \end{array} \]

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+282)
   (/
    (/
     (fma
      x
      (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
      z)
     (fma
      (fma x (+ x 43.3400022514) 263.505074721)
      (* x x)
      (fma x 313.399215894 47.066876606)))
    (/ 1.0 (+ x -2.0)))
   (*
    (+ x -2.0)
    (+
     4.16438922228
     (/
      (-
       (/ (+ 3451.550173699799 (/ (- y 124074.40615218398) x)) x)
       101.7851458539211)
      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+282) {
		tmp = (fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(fma(x, (x + 43.3400022514), 263.505074721), (x * x), fma(x, 313.399215894, 47.066876606))) / (1.0 / (x + -2.0));
	} else {
		tmp = (x + -2.0) * (4.16438922228 + ((((3451.550173699799 + ((y - 124074.40615218398) / x)) / x) - 101.7851458539211) / 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+282)
		tmp = Float64(Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(fma(x, Float64(x + 43.3400022514), 263.505074721), Float64(x * x), fma(x, 313.399215894, 47.066876606))) / Float64(1.0 / Float64(x + -2.0)));
	else
		tmp = Float64(Float64(x + -2.0) * Float64(4.16438922228 + Float64(Float64(Float64(Float64(3451.550173699799 + Float64(Float64(y - 124074.40615218398) / x)) / x) - 101.7851458539211) / 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+282], N[(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] / N[(1.0 / N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + -2.0), $MachinePrecision] * N[(4.16438922228 + N[(N[(N[(N[(3451.550173699799 + N[(N[(y - 124074.40615218398), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 101.7851458539211), $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))) < 1.00000000000000003e282

   1. Initial program 95.7%

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

   4. Applied rewrites 98.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-fma.f64 98.9

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

   6. Applied rewrites 98.9%

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

   if 1.00000000000000003e282 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 0.3

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

   5. Applied rewrites 0.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} + -1 \cdot \frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)\right)}\right) \cdot \left(x + -2\right) \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \color{blue}{\frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}}\right) \cdot \left(x + -2\right) \]

   10. Applied rewrites 99.1%

       \[\leadsto \color{blue}{\left(4.16438922228 - \frac{101.7851458539211 - \frac{3451.550173699799 - \frac{124074.40615218398 - y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+282}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), x \cdot x, \mathsf{fma}\left(x, 313.399215894, 47.066876606\right)\right)}}{\frac{1}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{\frac{3451.550173699799 + \frac{y - 124074.40615218398}{x}}{x} - 101.7851458539211}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+282}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{\frac{3451.550173699799 + \frac{y - 124074.40615218398}{x}}{x} - 101.7851458539211}{x}\right)\\ \end{array} \end{array} \]

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+282)
   (/
    (/
     (fma
      x
      (fma x (fma x (fma x 4.16438922228 78.6994924154) 137.519416416) y)
      z)
     (fma
      x
      (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
      47.066876606))
    (/ 1.0 (+ x -2.0)))
   (*
    (+ x -2.0)
    (+
     4.16438922228
     (/
      (-
       (/ (+ 3451.550173699799 (/ (- y 124074.40615218398) x)) x)
       101.7851458539211)
      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+282) {
		tmp = (fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)) / (1.0 / (x + -2.0));
	} else {
		tmp = (x + -2.0) * (4.16438922228 + ((((3451.550173699799 + ((y - 124074.40615218398) / x)) / x) - 101.7851458539211) / 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+282)
		tmp = Float64(Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)) / Float64(1.0 / Float64(x + -2.0)));
	else
		tmp = Float64(Float64(x + -2.0) * Float64(4.16438922228 + Float64(Float64(Float64(Float64(3451.550173699799 + Float64(Float64(y - 124074.40615218398) / x)) / x) - 101.7851458539211) / 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+282], N[(N[(N[(x * N[(x * N[(x * N[(x * 4.16438922228 + 78.6994924154), $MachinePrecision] + 137.519416416), $MachinePrecision] + y), $MachinePrecision] + z), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(x + -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + -2.0), $MachinePrecision] * N[(4.16438922228 + N[(N[(N[(N[(3451.550173699799 + N[(N[(y - 124074.40615218398), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 101.7851458539211), $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))) < 1.00000000000000003e282

   1. Initial program 95.7%

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

   4. Applied rewrites 98.8%

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

   if 1.00000000000000003e282 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 0.3

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

   5. Applied rewrites 0.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} + -1 \cdot \frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)\right)}\right) \cdot \left(x + -2\right) \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \color{blue}{\frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}}\right) \cdot \left(x + -2\right) \]

   10. Applied rewrites 99.1%

       \[\leadsto \color{blue}{\left(4.16438922228 - \frac{101.7851458539211 - \frac{3451.550173699799 - \frac{124074.40615218398 - y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+282}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}}{\frac{1}{x + -2}}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{\frac{3451.550173699799 + \frac{y - 124074.40615218398}{x}}{x} - 101.7851458539211}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+282}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{\frac{3451.550173699799 + \frac{y - 124074.40615218398}{x}}{x} - 101.7851458539211}{x}\right)\\ \end{array} \end{array} \]

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+282)
   (*
    (+ x -2.0)
    (/
     (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)
    (+
     4.16438922228
     (/
      (-
       (/ (+ 3451.550173699799 (/ (- y 124074.40615218398) x)) x)
       101.7851458539211)
      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+282) {
		tmp = (x + -2.0) * (fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606));
	} else {
		tmp = (x + -2.0) * (4.16438922228 + ((((3451.550173699799 + ((y - 124074.40615218398) / x)) / x) - 101.7851458539211) / 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+282)
		tmp = Float64(Float64(x + -2.0) * Float64(fma(x, fma(x, fma(x, fma(x, 4.16438922228, 78.6994924154), 137.519416416), y), z) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)));
	else
		tmp = Float64(Float64(x + -2.0) * Float64(4.16438922228 + Float64(Float64(Float64(Float64(3451.550173699799 + Float64(Float64(y - 124074.40615218398) / x)) / x) - 101.7851458539211) / 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+282], N[(N[(x + -2.0), $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[(N[(x + -2.0), $MachinePrecision] * N[(4.16438922228 + N[(N[(N[(N[(3451.550173699799 + N[(N[(y - 124074.40615218398), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 101.7851458539211), $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))) < 1.00000000000000003e282

   1. Initial program 95.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 98.8%

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

   if 1.00000000000000003e282 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.1%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 0.3

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

   5. Applied rewrites 0.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} + -1 \cdot \frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)\right)}\right) \cdot \left(x + -2\right) \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \color{blue}{\frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}}\right) \cdot \left(x + -2\right) \]

   10. Applied rewrites 99.1%

       \[\leadsto \color{blue}{\left(4.16438922228 - \frac{101.7851458539211 - \frac{3451.550173699799 - \frac{124074.40615218398 - y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot \left(x \cdot 4.16438922228 + 78.6994924154\right) + 137.519416416\right) + y\right) + z\right)}{x \cdot \left(x \cdot \left(x \cdot \left(x + 43.3400022514\right) + 263.505074721\right) + 313.399215894\right) + 47.066876606} \leq 10^{+282}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, 4.16438922228, 78.6994924154\right), 137.519416416\right), y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{\frac{3451.550173699799 + \frac{y - 124074.40615218398}{x}}{x} - 101.7851458539211}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -2\right) \cdot \left(4.16438922228 + \frac{\frac{3451.550173699799 + \frac{y - 124074.40615218398}{x}}{x} - 101.7851458539211}{x}\right)\\ \mathbf{if}\;x \leq -7100000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9500000:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          (+ x -2.0)
          (+
           4.16438922228
           (/
            (-
             (/ (+ 3451.550173699799 (/ (- y 124074.40615218398) x)) x)
             101.7851458539211)
            x)))))
   (if (<= x -7100000000000.0)
     t_0
     (if (<= x 9500000.0)
       (*
        (+ x -2.0)
        (/
         (fma x (fma x 137.519416416 y) z)
         (fma
          x
          (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
          47.066876606)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + -2.0) * (4.16438922228 + ((((3451.550173699799 + ((y - 124074.40615218398) / x)) / x) - 101.7851458539211) / x));
	double tmp;
	if (x <= -7100000000000.0) {
		tmp = t_0;
	} else if (x <= 9500000.0) {
		tmp = (x + -2.0) * (fma(x, fma(x, 137.519416416, y), z) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + -2.0) * Float64(4.16438922228 + Float64(Float64(Float64(Float64(3451.550173699799 + Float64(Float64(y - 124074.40615218398) / x)) / x) - 101.7851458539211) / x)))
	tmp = 0.0
	if (x <= -7100000000000.0)
		tmp = t_0;
	elseif (x <= 9500000.0)
		tmp = Float64(Float64(x + -2.0) * Float64(fma(x, fma(x, 137.519416416, y), z) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + -2.0), $MachinePrecision] * N[(4.16438922228 + N[(N[(N[(N[(3451.550173699799 + N[(N[(y - 124074.40615218398), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - 101.7851458539211), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7100000000000.0], t$95$0, If[LessEqual[x, 9500000.0], N[(N[(x + -2.0), $MachinePrecision] * N[(N[(x * N[(x * 137.519416416 + 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.1e12 or 9.5e6 < x

   1. Initial program 14.8%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 5.0

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

   5. Applied rewrites 5.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 8.0%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} + -1 \cdot \frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)\right)}\right) \cdot \left(x + -2\right) \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \color{blue}{\frac{\frac{12723143231740136880149}{125000000000000000000} + -1 \cdot \frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} + -1 \cdot \frac{\frac{387732519225574910908939577061312055388407301}{3125000000000000000000000000000000000000} + -1 \cdot y}{x}}{x}}{x}}\right) \cdot \left(x + -2\right) \]

   10. Applied rewrites 95.5%

       \[\leadsto \color{blue}{\left(4.16438922228 - \frac{101.7851458539211 - \frac{3451.550173699799 - \frac{124074.40615218398 - y}{x}}{x}}{x}\right)} \cdot \left(x + -2\right) \]

   if -7.1e12 < x < 9.5e6

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 99.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7100000000000:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{\frac{3451.550173699799 + \frac{y - 124074.40615218398}{x}}{x} - 101.7851458539211}{x}\right)\\ \mathbf{elif}\;x \leq 9500000:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{\frac{3451.550173699799 + \frac{y - 124074.40615218398}{x}}{x} - 101.7851458539211}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -32000000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+35}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -32000000000000.0)
   (* x (+ 4.16438922228 (/ -110.1139242984811 x)))
   (if (<= x 3.5e+35)
     (*
      (+ x -2.0)
      (/
       (fma x (fma x 137.519416416 y) z)
       (fma
        x
        (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
        47.066876606)))
     (* x 4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -32000000000000.0) {
		tmp = x * (4.16438922228 + (-110.1139242984811 / x));
	} else if (x <= 3.5e+35) {
		tmp = (x + -2.0) * (fma(x, fma(x, 137.519416416, y), z) / fma(x, fma(x, fma(x, (x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606));
	} else {
		tmp = x * 4.16438922228;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -32000000000000.0)
		tmp = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)));
	elseif (x <= 3.5e+35)
		tmp = Float64(Float64(x + -2.0) * Float64(fma(x, fma(x, 137.519416416, y), z) / fma(x, fma(x, fma(x, Float64(x + 43.3400022514), 263.505074721), 313.399215894), 47.066876606)));
	else
		tmp = Float64(x * 4.16438922228);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -32000000000000.0], N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e+35], N[(N[(x + -2.0), $MachinePrecision] * N[(N[(x * N[(x * 137.519416416 + 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 * 4.16438922228), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2e13

   1. Initial program 13.4%

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

   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 90.0

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

   5. Applied rewrites 90.0%

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

   if -3.2e13 < x < 3.5000000000000001e35

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 96.4

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

   5. Applied rewrites 96.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 96.4%

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

   if 3.5000000000000001e35 < x

   1. Initial program 5.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -32000000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+35}:\\ \;\;\;\;\left(x + -2\right) \cdot \frac{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -32000000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right) \cdot \frac{x + -2}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x + 43.3400022514, 263.505074721\right), 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 4.16438922228\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -32000000000000.0)
   (* x (+ 4.16438922228 (/ -110.1139242984811 x)))
   (if (<= x 3.5e+35)
     (*
      (fma x (fma x 137.519416416 y) z)
      (/
       (+ x -2.0)
       (fma
        x
        (fma x (fma x (+ x 43.3400022514) 263.505074721) 313.399215894)
        47.066876606)))
     (* x 4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -32000000000000.0) {
		tmp = x * (4.16438922228 + (-110.1139242984811 / x));
	} else if (x <= 3.5e+35) {
		tmp = fma(x, fma(x, 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 * 4.16438922228;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -32000000000000.0)
		tmp = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)));
	elseif (x <= 3.5e+35)
		tmp = Float64(fma(x, fma(x, 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 * 4.16438922228);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -32000000000000.0], N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.5e+35], N[(N[(x * N[(x * 137.519416416 + y), $MachinePrecision] + z), $MachinePrecision] * N[(N[(x + -2.0), $MachinePrecision] / N[(x * N[(x * N[(x * N[(x + 43.3400022514), $MachinePrecision] + 263.505074721), $MachinePrecision] + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * 4.16438922228), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2e13

   1. Initial program 13.4%

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

   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 90.0

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

   5. Applied rewrites 90.0%

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

   if -3.2e13 < x < 3.5000000000000001e35

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 96.4

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

   5. Applied rewrites 96.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-+.f64 N/A

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

   7. Applied rewrites 96.2%

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

   if 3.5000000000000001e35 < x

   1. Initial program 5.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 94.3

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

   5. Applied rewrites 94.3%

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

Alternative 7: 92.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 1450:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 263.505074721, 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211 + \frac{-3451.550173699799}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -43000000000.0)
   (* x (+ 4.16438922228 (/ -110.1139242984811 x)))
   (if (<= x 1450.0)
     (/
      (* (- x 2.0) (fma x (fma x 137.519416416 y) z))
      (fma x (fma x 263.505074721 313.399215894) 47.066876606))
     (*
      (+ x -2.0)
      (-
       4.16438922228
       (/ (+ 101.7851458539211 (/ -3451.550173699799 x)) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -43000000000.0) {
		tmp = x * (4.16438922228 + (-110.1139242984811 / x));
	} else if (x <= 1450.0) {
		tmp = ((x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / fma(x, fma(x, 263.505074721, 313.399215894), 47.066876606);
	} else {
		tmp = (x + -2.0) * (4.16438922228 - ((101.7851458539211 + (-3451.550173699799 / x)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)));
	elseif (x <= 1450.0)
		tmp = Float64(Float64(Float64(x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / fma(x, fma(x, 263.505074721, 313.399215894), 47.066876606));
	else
		tmp = Float64(Float64(x + -2.0) * Float64(4.16438922228 - Float64(Float64(101.7851458539211 + Float64(-3451.550173699799 / x)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -43000000000.0], N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1450.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(x * N[(x * 137.519416416 + y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x * 263.505074721 + 313.399215894), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], N[(N[(x + -2.0), $MachinePrecision] * N[(4.16438922228 - N[(N[(101.7851458539211 + N[(-3451.550173699799 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e10

   1. Initial program 13.4%

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

   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 90.0

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

   5. Applied rewrites 90.0%

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

   if -4.3e10 < x < 1450

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 96.1

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

   8. Applied rewrites 96.1%

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

   if 1450 < x

   1. Initial program 15.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 8.2

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

   5. Applied rewrites 8.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 9.5%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} + -1 \cdot \frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}\right)} \cdot \left(x + -2\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}\right)\right)}\right) \cdot \left(x + -2\right) \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \color{blue}{\frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}}\right) \cdot \left(x + -2\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\color{blue}{\frac{12723143231740136880149}{125000000000000000000} + \left(\mathsf{neg}\left(\frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}\right)\right)}}{x}\right) \cdot \left(x + -2\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\color{blue}{\frac{12723143231740136880149}{125000000000000000000} + \left(\mathsf{neg}\left(\frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}\right)\right)}}{x}\right) \cdot \left(x + -2\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot 1}{x}}\right)\right)}{x}\right) \cdot \left(x + -2\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2157218858562374472887084159837293}{625000000000000000000000000000}}}{x}\right)\right)}{x}\right) \cdot \left(x + -2\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2157218858562374472887084159837293}{625000000000000000000000000000}\right)}{x}}}{x}\right) \cdot \left(x + -2\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2157218858562374472887084159837293}{625000000000000000000000000000}\right)}{x}}}{x}\right) \cdot \left(x + -2\right) \]

      11. metadata-eval 89.3

          \[\leadsto \left(4.16438922228 - \frac{101.7851458539211 + \frac{\color{blue}{-3451.550173699799}}{x}}{x}\right) \cdot \left(x + -2\right) \]

   10. Applied rewrites 89.3%

       \[\leadsto \color{blue}{\left(4.16438922228 - \frac{101.7851458539211 + \frac{-3451.550173699799}{x}}{x}\right)} \cdot \left(x + -2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 1450:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 263.505074721, 313.399215894\right), 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211 + \frac{-3451.550173699799}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211 + \frac{-3451.550173699799}{x}}{x}\right)\\ \mathbf{if}\;x \leq -36:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 460:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{\mathsf{fma}\left(x, 313.399215894, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          (+ x -2.0)
          (-
           4.16438922228
           (/ (+ 101.7851458539211 (/ -3451.550173699799 x)) x)))))
   (if (<= x -36.0)
     t_0
     (if (<= x 460.0)
       (/
        (* (- x 2.0) (fma x (fma x 137.519416416 y) z))
        (fma x 313.399215894 47.066876606))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + -2.0) * (4.16438922228 - ((101.7851458539211 + (-3451.550173699799 / x)) / x));
	double tmp;
	if (x <= -36.0) {
		tmp = t_0;
	} else if (x <= 460.0) {
		tmp = ((x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / fma(x, 313.399215894, 47.066876606);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + -2.0) * Float64(4.16438922228 - Float64(Float64(101.7851458539211 + Float64(-3451.550173699799 / x)) / x)))
	tmp = 0.0
	if (x <= -36.0)
		tmp = t_0;
	elseif (x <= 460.0)
		tmp = Float64(Float64(Float64(x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / fma(x, 313.399215894, 47.066876606));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + -2.0), $MachinePrecision] * N[(4.16438922228 - N[(N[(101.7851458539211 + N[(-3451.550173699799 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -36.0], t$95$0, If[LessEqual[x, 460.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(x * N[(x * 137.519416416 + y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(x * 313.399215894 + 47.066876606), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -36 or 460 < x

   1. Initial program 16.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 6.4

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

   5. Applied rewrites 6.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 9.4%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} + -1 \cdot \frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}\right)} \cdot \left(x + -2\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}\right)\right)}\right) \cdot \left(x + -2\right) \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \color{blue}{\frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}}\right) \cdot \left(x + -2\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\color{blue}{\frac{12723143231740136880149}{125000000000000000000} + \left(\mathsf{neg}\left(\frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}\right)\right)}}{x}\right) \cdot \left(x + -2\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\color{blue}{\frac{12723143231740136880149}{125000000000000000000} + \left(\mathsf{neg}\left(\frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}\right)\right)}}{x}\right) \cdot \left(x + -2\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot 1}{x}}\right)\right)}{x}\right) \cdot \left(x + -2\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2157218858562374472887084159837293}{625000000000000000000000000000}}}{x}\right)\right)}{x}\right) \cdot \left(x + -2\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2157218858562374472887084159837293}{625000000000000000000000000000}\right)}{x}}}{x}\right) \cdot \left(x + -2\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2157218858562374472887084159837293}{625000000000000000000000000000}\right)}{x}}}{x}\right) \cdot \left(x + -2\right) \]

      11. metadata-eval 88.3

          \[\leadsto \left(4.16438922228 - \frac{101.7851458539211 + \frac{\color{blue}{-3451.550173699799}}{x}}{x}\right) \cdot \left(x + -2\right) \]

   10. Applied rewrites 88.3%

       \[\leadsto \color{blue}{\left(4.16438922228 - \frac{101.7851458539211 + \frac{-3451.550173699799}{x}}{x}\right)} \cdot \left(x + -2\right) \]

   if -36 < x < 460

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.0

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

   8. Applied rewrites 97.0%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -36:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211 + \frac{-3451.550173699799}{x}}{x}\right)\\ \mathbf{elif}\;x \leq 460:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{\mathsf{fma}\left(x, 313.399215894, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211 + \frac{-3451.550173699799}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 460:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211 + \frac{-3451.550173699799}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -43000000000.0)
   (* x (+ 4.16438922228 (/ -110.1139242984811 x)))
   (if (<= x 460.0)
     (/ (* (- x 2.0) (fma x (fma x 137.519416416 y) z)) 47.066876606)
     (*
      (+ x -2.0)
      (-
       4.16438922228
       (/ (+ 101.7851458539211 (/ -3451.550173699799 x)) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -43000000000.0) {
		tmp = x * (4.16438922228 + (-110.1139242984811 / x));
	} else if (x <= 460.0) {
		tmp = ((x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / 47.066876606;
	} else {
		tmp = (x + -2.0) * (4.16438922228 - ((101.7851458539211 + (-3451.550173699799 / x)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)));
	elseif (x <= 460.0)
		tmp = Float64(Float64(Float64(x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / 47.066876606);
	else
		tmp = Float64(Float64(x + -2.0) * Float64(4.16438922228 - Float64(Float64(101.7851458539211 + Float64(-3451.550173699799 / x)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -43000000000.0], N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 460.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(x * N[(x * 137.519416416 + y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / 47.066876606), $MachinePrecision], N[(N[(x + -2.0), $MachinePrecision] * N[(4.16438922228 - N[(N[(101.7851458539211 + N[(-3451.550173699799 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e10

   1. Initial program 13.4%

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

   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 90.0

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

   5. Applied rewrites 90.0%

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

   if -4.3e10 < x < 460

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4297481763}{31250000}, y\right), z\right)}{\color{blue}{\frac{23533438303}{500000000}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.3%

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

   if 460 < x

   1. Initial program 15.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 8.2

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

   5. Applied rewrites 8.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 9.5%

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

   8. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} + -1 \cdot \frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}\right)} \cdot \left(x + -2\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}\right)\right)}\right) \cdot \left(x + -2\right) \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}\right)} \cdot \left(x + -2\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \color{blue}{\frac{\frac{12723143231740136880149}{125000000000000000000} - \frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}}{x}}\right) \cdot \left(x + -2\right) \]

      5. sub-neg N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\color{blue}{\frac{12723143231740136880149}{125000000000000000000} + \left(\mathsf{neg}\left(\frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}\right)\right)}}{x}\right) \cdot \left(x + -2\right) \]

      6. lower-+.f64 N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\color{blue}{\frac{12723143231740136880149}{125000000000000000000} + \left(\mathsf{neg}\left(\frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot \frac{1}{x}\right)\right)}}{x}\right) \cdot \left(x + -2\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{2157218858562374472887084159837293}{625000000000000000000000000000} \cdot 1}{x}}\right)\right)}{x}\right) \cdot \left(x + -2\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2157218858562374472887084159837293}{625000000000000000000000000000}}}{x}\right)\right)}{x}\right) \cdot \left(x + -2\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2157218858562374472887084159837293}{625000000000000000000000000000}\right)}{x}}}{x}\right) \cdot \left(x + -2\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\frac{104109730557}{25000000000} - \frac{\frac{12723143231740136880149}{125000000000000000000} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2157218858562374472887084159837293}{625000000000000000000000000000}\right)}{x}}}{x}\right) \cdot \left(x + -2\right) \]

      11. metadata-eval 89.3

          \[\leadsto \left(4.16438922228 - \frac{101.7851458539211 + \frac{\color{blue}{-3451.550173699799}}{x}}{x}\right) \cdot \left(x + -2\right) \]

   10. Applied rewrites 89.3%

       \[\leadsto \color{blue}{\left(4.16438922228 - \frac{101.7851458539211 + \frac{-3451.550173699799}{x}}{x}\right)} \cdot \left(x + -2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 460:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 - \frac{101.7851458539211 + \frac{-3451.550173699799}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 460:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -43000000000.0)
   (* x (+ 4.16438922228 (/ -110.1139242984811 x)))
   (if (<= x 460.0)
     (/ (* (- x 2.0) (fma x (fma x 137.519416416 y) z)) 47.066876606)
     (*
      x
      (+
       4.16438922228
       (/ (+ -110.1139242984811 (/ 3655.1204654076414 x)) x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -43000000000.0) {
		tmp = x * (4.16438922228 + (-110.1139242984811 / x));
	} else if (x <= 460.0) {
		tmp = ((x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / 47.066876606;
	} else {
		tmp = x * (4.16438922228 + ((-110.1139242984811 + (3655.1204654076414 / x)) / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)));
	elseif (x <= 460.0)
		tmp = Float64(Float64(Float64(x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / 47.066876606);
	else
		tmp = Float64(x * Float64(4.16438922228 + Float64(Float64(-110.1139242984811 + Float64(3655.1204654076414 / x)) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -43000000000.0], N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 460.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(x * N[(x * 137.519416416 + y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / 47.066876606), $MachinePrecision], N[(x * N[(4.16438922228 + N[(N[(-110.1139242984811 + N[(3655.1204654076414 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e10

   1. Initial program 13.4%

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

   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 90.0

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

   5. Applied rewrites 90.0%

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

   if -4.3e10 < x < 460

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4297481763}{31250000}, y\right), z\right)}{\color{blue}{\frac{23533438303}{500000000}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.3%

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

   if 460 < x

   1. Initial program 15.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate--l+ N/A

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

      3. lower-+.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. div-sub N/A

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

      11. lower-/.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-+.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 460:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{47.066876606}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811 + \frac{3655.1204654076414}{x}}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 460:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{-101.7851458539211}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -43000000000.0)
   (* x (+ 4.16438922228 (/ -110.1139242984811 x)))
   (if (<= x 460.0)
     (/ (* (- x 2.0) (fma x (fma x 137.519416416 y) z)) 47.066876606)
     (* (+ x -2.0) (+ 4.16438922228 (/ -101.7851458539211 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -43000000000.0) {
		tmp = x * (4.16438922228 + (-110.1139242984811 / x));
	} else if (x <= 460.0) {
		tmp = ((x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / 47.066876606;
	} else {
		tmp = (x + -2.0) * (4.16438922228 + (-101.7851458539211 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)));
	elseif (x <= 460.0)
		tmp = Float64(Float64(Float64(x - 2.0) * fma(x, fma(x, 137.519416416, y), z)) / 47.066876606);
	else
		tmp = Float64(Float64(x + -2.0) * Float64(4.16438922228 + Float64(-101.7851458539211 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -43000000000.0], N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 460.0], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(x * N[(x * 137.519416416 + y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / 47.066876606), $MachinePrecision], N[(N[(x + -2.0), $MachinePrecision] * N[(4.16438922228 + N[(-101.7851458539211 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e10

   1. Initial program 13.4%

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

   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 90.0

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

   5. Applied rewrites 90.0%

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

   if -4.3e10 < x < 460

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{4297481763}{31250000}, y\right), z\right)}{\color{blue}{\frac{23533438303}{500000000}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.3%

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

   if 460 < x

   1. Initial program 15.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 8.2

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

   5. Applied rewrites 8.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 9.5%

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

   8. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval 89.1

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

   10. Applied rewrites 89.1%

       \[\leadsto \color{blue}{\left(4.16438922228 + \frac{-101.7851458539211}{x}\right)} \cdot \left(x + -2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 460:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 137.519416416, y\right), z\right)}{47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{-101.7851458539211}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 24:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, z \cdot 0.3041881842569256\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{-101.7851458539211}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -43000000000.0)
   (* x (+ 4.16438922228 (/ -110.1139242984811 x)))
   (if (<= x 24.0)
     (fma
      x
      (fma -0.0424927283095952 y (* z 0.3041881842569256))
      (* z -0.0424927283095952))
     (* (+ x -2.0) (+ 4.16438922228 (/ -101.7851458539211 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -43000000000.0) {
		tmp = x * (4.16438922228 + (-110.1139242984811 / x));
	} else if (x <= 24.0) {
		tmp = fma(x, fma(-0.0424927283095952, y, (z * 0.3041881842569256)), (z * -0.0424927283095952));
	} else {
		tmp = (x + -2.0) * (4.16438922228 + (-101.7851458539211 / x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)));
	elseif (x <= 24.0)
		tmp = fma(x, fma(-0.0424927283095952, y, Float64(z * 0.3041881842569256)), Float64(z * -0.0424927283095952));
	else
		tmp = Float64(Float64(x + -2.0) * Float64(4.16438922228 + Float64(-101.7851458539211 / x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -43000000000.0], N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 24.0], N[(x * N[(-0.0424927283095952 * y + N[(z * 0.3041881842569256), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], N[(N[(x + -2.0), $MachinePrecision] * N[(4.16438922228 + N[(-101.7851458539211 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.3e10

   1. Initial program 13.4%

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

   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 90.0

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

   5. Applied rewrites 90.0%

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

   if -4.3e10 < x < 24

   1. Initial program 99.6%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 89.9

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

   7. Applied rewrites 89.9%

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

   if 24 < x

   1. Initial program 15.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 8.2

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

   5. Applied rewrites 8.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 9.5%

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

   8. Taylor expanded in x around inf

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

      1. sub-neg N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval 89.1

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

   10. Applied rewrites 89.1%

       \[\leadsto \color{blue}{\left(4.16438922228 + \frac{-101.7851458539211}{x}\right)} \cdot \left(x + -2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 24:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, z \cdot 0.3041881842569256\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(4.16438922228 + \frac{-101.7851458539211}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 28:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, z \cdot 0.3041881842569256\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ 4.16438922228 (/ -110.1139242984811 x)))))
   (if (<= x -43000000000.0)
     t_0
     (if (<= x 28.0)
       (fma
        x
        (fma -0.0424927283095952 y (* z 0.3041881842569256))
        (* z -0.0424927283095952))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.16438922228 + (-110.1139242984811 / x));
	double tmp;
	if (x <= -43000000000.0) {
		tmp = t_0;
	} else if (x <= 28.0) {
		tmp = fma(x, fma(-0.0424927283095952, y, (z * 0.3041881842569256)), (z * -0.0424927283095952));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)))
	tmp = 0.0
	if (x <= -43000000000.0)
		tmp = t_0;
	elseif (x <= 28.0)
		tmp = fma(x, fma(-0.0424927283095952, y, Float64(z * 0.3041881842569256)), Float64(z * -0.0424927283095952));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -43000000000.0], t$95$0, If[LessEqual[x, 28.0], N[(x * N[(-0.0424927283095952 * y + N[(z * 0.3041881842569256), $MachinePrecision]), $MachinePrecision] + N[(z * -0.0424927283095952), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3e10 or 28 < x

   1. Initial program 14.8%

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

   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 89.5

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

   5. Applied rewrites 89.5%

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

   if -4.3e10 < x < 28

   1. Initial program 99.6%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 89.9

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

   7. Applied rewrites 89.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -43000000000:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{elif}\;x \leq 28:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(-0.0424927283095952, y, z \cdot 0.3041881842569256\right), z \cdot -0.0424927283095952\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-23}:\\ \;\;\;\;\left(x + -2\right) \cdot \left(z \cdot \mathsf{fma}\left(x, -0.14147091005106402, 0.0212463641547976\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ 4.16438922228 (/ -110.1139242984811 x)))))
   (if (<= x -0.029)
     t_0
     (if (<= x 5.8e-23)
       (* (+ x -2.0) (* z (fma x -0.14147091005106402 0.0212463641547976)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.16438922228 + (-110.1139242984811 / x));
	double tmp;
	if (x <= -0.029) {
		tmp = t_0;
	} else if (x <= 5.8e-23) {
		tmp = (x + -2.0) * (z * fma(x, -0.14147091005106402, 0.0212463641547976));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)))
	tmp = 0.0
	if (x <= -0.029)
		tmp = t_0;
	elseif (x <= 5.8e-23)
		tmp = Float64(Float64(x + -2.0) * Float64(z * fma(x, -0.14147091005106402, 0.0212463641547976)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.029], t$95$0, If[LessEqual[x, 5.8e-23], N[(N[(x + -2.0), $MachinePrecision] * N[(z * N[(x * -0.14147091005106402 + 0.0212463641547976), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.1%

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

   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 85.2

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

   5. Applied rewrites 85.2%

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

   if -0.0290000000000000015 < x < 5.8000000000000003e-23

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 72.3

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

   5. Applied rewrites 72.3%

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

   7. Applied rewrites 72.2%

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

   8. Taylor expanded in x around 0

      \[\leadsto \left(x + -2\right) \cdot \left(z \cdot \left(\frac{500000000}{23533438303} + \color{blue}{\frac{-78349803973500000000}{553822718361107519809} \cdot x}\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 71.9%

       \[\leadsto \left(x + -2\right) \cdot \left(z \cdot \mathsf{fma}\left(x, \color{blue}{-0.14147091005106402}, 0.0212463641547976\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 76.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(4.16438922228 + \frac{-110.1139242984811}{x}\right)\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(-0.0424927283095952, z, x \cdot \left(z \cdot 0.3041881842569256\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (+ 4.16438922228 (/ -110.1139242984811 x)))))
   (if (<= x -0.029)
     t_0
     (if (<= x 5.8e-23)
       (fma -0.0424927283095952 z (* x (* z 0.3041881842569256)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.16438922228 + (-110.1139242984811 / x));
	double tmp;
	if (x <= -0.029) {
		tmp = t_0;
	} else if (x <= 5.8e-23) {
		tmp = fma(-0.0424927283095952, z, (x * (z * 0.3041881842569256)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.16438922228 + Float64(-110.1139242984811 / x)))
	tmp = 0.0
	if (x <= -0.029)
		tmp = t_0;
	elseif (x <= 5.8e-23)
		tmp = fma(-0.0424927283095952, z, Float64(x * Float64(z * 0.3041881842569256)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.16438922228 + N[(-110.1139242984811 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.029], t$95$0, If[LessEqual[x, 5.8e-23], N[(-0.0424927283095952 * z + N[(x * N[(z * 0.3041881842569256), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.1%

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

   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 85.2

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

   5. Applied rewrites 85.2%

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

   if -0.0290000000000000015 < x < 5.8000000000000003e-23

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.9%

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

Alternative 16: 76.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -2\right) \cdot 4.16438922228\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(-0.0424927283095952, z, x \cdot \left(z \cdot 0.3041881842569256\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ x -2.0) 4.16438922228)))
   (if (<= x -0.029)
     t_0
     (if (<= x 5.8e-23)
       (fma -0.0424927283095952 z (* x (* z 0.3041881842569256)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + -2.0) * 4.16438922228;
	double tmp;
	if (x <= -0.029) {
		tmp = t_0;
	} else if (x <= 5.8e-23) {
		tmp = fma(-0.0424927283095952, z, (x * (z * 0.3041881842569256)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + -2.0) * 4.16438922228)
	tmp = 0.0
	if (x <= -0.029)
		tmp = t_0;
	elseif (x <= 5.8e-23)
		tmp = fma(-0.0424927283095952, z, Float64(x * Float64(z * 0.3041881842569256)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + -2.0), $MachinePrecision] * 4.16438922228), $MachinePrecision]}, If[LessEqual[x, -0.029], t$95$0, If[LessEqual[x, 5.8e-23], N[(-0.0424927283095952 * z + N[(x * N[(z * 0.3041881842569256), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.1%

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

   4. Applied rewrites 23.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-fma.f64 23.2

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

   6. Applied rewrites 23.2%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{104109730557}{25000000000}}}{\frac{1}{x + -2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 84.4%

      \[\leadsto \frac{\color{blue}{4.16438922228}}{\frac{1}{x + -2}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{104109730557}{25000000000}}{\frac{1}{x + -2}}} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{104109730557}{25000000000}}{\color{blue}{\frac{1}{x + -2}}} \]

       3. associate-/r/ N/A

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

       4. /-rgt-identity N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 84.4

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

   11. Applied rewrites 84.4%

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

   if -0.0290000000000000015 < x < 5.8000000000000003e-23

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.9%

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

Alternative 17: 76.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -2\right) \cdot 4.16438922228\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-23}:\\ \;\;\;\;z \cdot \left(\left(x + -2\right) \cdot 0.0212463641547976\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ x -2.0) 4.16438922228)))
   (if (<= x -0.029)
     t_0
     (if (<= x 5.8e-23) (* z (* (+ x -2.0) 0.0212463641547976)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + -2.0) * 4.16438922228;
	double tmp;
	if (x <= -0.029) {
		tmp = t_0;
	} else if (x <= 5.8e-23) {
		tmp = z * ((x + -2.0) * 0.0212463641547976);
	} 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 = (x + (-2.0d0)) * 4.16438922228d0
    if (x <= (-0.029d0)) then
        tmp = t_0
    else if (x <= 5.8d-23) then
        tmp = z * ((x + (-2.0d0)) * 0.0212463641547976d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + -2.0) * 4.16438922228;
	double tmp;
	if (x <= -0.029) {
		tmp = t_0;
	} else if (x <= 5.8e-23) {
		tmp = z * ((x + -2.0) * 0.0212463641547976);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + -2.0) * 4.16438922228
	tmp = 0
	if x <= -0.029:
		tmp = t_0
	elif x <= 5.8e-23:
		tmp = z * ((x + -2.0) * 0.0212463641547976)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + -2.0) * 4.16438922228)
	tmp = 0.0
	if (x <= -0.029)
		tmp = t_0;
	elseif (x <= 5.8e-23)
		tmp = Float64(z * Float64(Float64(x + -2.0) * 0.0212463641547976));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + -2.0) * 4.16438922228;
	tmp = 0.0;
	if (x <= -0.029)
		tmp = t_0;
	elseif (x <= 5.8e-23)
		tmp = z * ((x + -2.0) * 0.0212463641547976);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + -2.0), $MachinePrecision] * 4.16438922228), $MachinePrecision]}, If[LessEqual[x, -0.029], t$95$0, If[LessEqual[x, 5.8e-23], N[(z * N[(N[(x + -2.0), $MachinePrecision] * 0.0212463641547976), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.1%

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

   4. Applied rewrites 23.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-fma.f64 23.2

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

   6. Applied rewrites 23.2%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{104109730557}{25000000000}}}{\frac{1}{x + -2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 84.4%

      \[\leadsto \frac{\color{blue}{4.16438922228}}{\frac{1}{x + -2}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{104109730557}{25000000000}}{\frac{1}{x + -2}}} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{104109730557}{25000000000}}{\color{blue}{\frac{1}{x + -2}}} \]

       3. associate-/r/ N/A

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

       4. /-rgt-identity N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 84.4

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

   11. Applied rewrites 84.4%

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

   if -0.0290000000000000015 < x < 5.8000000000000003e-23

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 72.3

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

   5. Applied rewrites 72.3%

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

   7. Applied rewrites 72.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 71.7%

       \[\leadsto \left(x + -2\right) \cdot \left(z \cdot 0.0212463641547976\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 71.7%

       \[\leadsto \left(\left(x + -2\right) \cdot 0.0212463641547976\right) \cdot \color{blue}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;\left(x + -2\right) \cdot 4.16438922228\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-23}:\\ \;\;\;\;z \cdot \left(\left(x + -2\right) \cdot 0.0212463641547976\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x + -2\right) \cdot 4.16438922228\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 76.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x + -2\right) \cdot 4.16438922228\\ \mathbf{if}\;x \leq -0.029:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-23}:\\ \;\;\;\;z \cdot -0.0424927283095952\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ x -2.0) 4.16438922228)))
   (if (<= x -0.029) t_0 (if (<= x 5.8e-23) (* z -0.0424927283095952) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x + -2.0) * 4.16438922228;
	double tmp;
	if (x <= -0.029) {
		tmp = t_0;
	} else if (x <= 5.8e-23) {
		tmp = z * -0.0424927283095952;
	} 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 = (x + (-2.0d0)) * 4.16438922228d0
    if (x <= (-0.029d0)) then
        tmp = t_0
    else if (x <= 5.8d-23) then
        tmp = z * (-0.0424927283095952d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x + -2.0) * 4.16438922228;
	double tmp;
	if (x <= -0.029) {
		tmp = t_0;
	} else if (x <= 5.8e-23) {
		tmp = z * -0.0424927283095952;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x + -2.0) * 4.16438922228
	tmp = 0
	if x <= -0.029:
		tmp = t_0
	elif x <= 5.8e-23:
		tmp = z * -0.0424927283095952
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x + -2.0) * 4.16438922228)
	tmp = 0.0
	if (x <= -0.029)
		tmp = t_0;
	elseif (x <= 5.8e-23)
		tmp = Float64(z * -0.0424927283095952);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x + -2.0) * 4.16438922228;
	tmp = 0.0;
	if (x <= -0.029)
		tmp = t_0;
	elseif (x <= 5.8e-23)
		tmp = z * -0.0424927283095952;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + -2.0), $MachinePrecision] * 4.16438922228), $MachinePrecision]}, If[LessEqual[x, -0.029], t$95$0, If[LessEqual[x, 5.8e-23], N[(z * -0.0424927283095952), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 19.1%

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

   4. Applied rewrites 23.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-fma.f64 23.2

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

   6. Applied rewrites 23.2%

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

   7. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{104109730557}{25000000000}}}{\frac{1}{x + -2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 84.4%

      \[\leadsto \frac{\color{blue}{4.16438922228}}{\frac{1}{x + -2}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{104109730557}{25000000000}}{\frac{1}{x + -2}}} \]

       2. lift-/.f64 N/A

          \[\leadsto \frac{\frac{104109730557}{25000000000}}{\color{blue}{\frac{1}{x + -2}}} \]

       3. associate-/r/ N/A

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

       4. /-rgt-identity N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 84.4

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

   11. Applied rewrites 84.4%

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

   if -0.0290000000000000015 < x < 5.8000000000000003e-23

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

Alternative 19: 76.3% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-0.029d0)) then
        tmp = x * 4.16438922228d0
    else if (x <= 0.4d0) then
        tmp = z * (-0.0424927283095952d0)
    else
        tmp = x * 4.16438922228d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -0.0290000000000000015 or 0.40000000000000002 < x

   1. Initial program 17.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   if -0.0290000000000000015 < x < 0.40000000000000002

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

Alternative 20: 45.7% accurate, 13.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 46.6

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

5. Applied rewrites 46.6%

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

Developer Target 1: 98.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{if}\;x < -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))
   (if (< x -3.326128725870005e+62)
     t_0
     (if (< x 9.429991714554673e+55)
       (*
        (/ (- x 2.0) 1.0)
        (/
         (+
          (*
           (+
            (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x)
            y)
           x)
          z)
         (+
          (*
           (+
            (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x))))
            313.399215894)
           x)
          47.066876606)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / (x * x)) + (4.16438922228 * x)) - 110.1139242984811;
	double tmp;
	if (x < -3.326128725870005e+62) {
		tmp = t_0;
	} else if (x < 9.429991714554673e+55) {
		tmp = ((x - 2.0) / 1.0) * (((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z) / (((((263.505074721 * x) + ((43.3400022514 * (x * x)) + (x * (x * x)))) + 313.399215894) * x) + 47.066876606));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / (x * x)) + (4.16438922228d0 * x)) - 110.1139242984811d0
    if (x < (-3.326128725870005d+62)) then
        tmp = t_0
    else if (x < 9.429991714554673d+55) then
        tmp = ((x - 2.0d0) / 1.0d0) * (((((((((x * 4.16438922228d0) + 78.6994924154d0) * x) + 137.519416416d0) * x) + y) * x) + z) / (((((263.505074721d0 * x) + ((43.3400022514d0 * (x * x)) + (x * (x * x)))) + 313.399215894d0) * x) + 47.066876606d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / (x * x)) + (4.16438922228 * x)) - 110.1139242984811;
	double tmp;
	if (x < -3.326128725870005e+62) {
		tmp = t_0;
	} else if (x < 9.429991714554673e+55) {
		tmp = ((x - 2.0) / 1.0) * (((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z) / (((((263.505074721 * x) + ((43.3400022514 * (x * x)) + (x * (x * x)))) + 313.399215894) * x) + 47.066876606));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / (x * x)) + (4.16438922228 * x)) - 110.1139242984811
	tmp = 0
	if x < -3.326128725870005e+62:
		tmp = t_0
	elif x < 9.429991714554673e+55:
		tmp = ((x - 2.0) / 1.0) * (((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z) / (((((263.505074721 * x) + ((43.3400022514 * (x * x)) + (x * (x * x)))) + 313.399215894) * x) + 47.066876606))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / Float64(x * x)) + Float64(4.16438922228 * x)) - 110.1139242984811)
	tmp = 0.0
	if (x < -3.326128725870005e+62)
		tmp = t_0;
	elseif (x < 9.429991714554673e+55)
		tmp = Float64(Float64(Float64(x - 2.0) / 1.0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z) / Float64(Float64(Float64(Float64(Float64(263.505074721 * x) + Float64(Float64(43.3400022514 * Float64(x * x)) + Float64(x * Float64(x * x)))) + 313.399215894) * x) + 47.066876606)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / (x * x)) + (4.16438922228 * x)) - 110.1139242984811;
	tmp = 0.0;
	if (x < -3.326128725870005e+62)
		tmp = t_0;
	elseif (x < 9.429991714554673e+55)
		tmp = ((x - 2.0) / 1.0) * (((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z) / (((((263.505074721 * x) + ((43.3400022514 * (x * x)) + (x * (x * x)))) + 313.399215894) * x) + 47.066876606));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(4.16438922228 * x), $MachinePrecision]), $MachinePrecision] - 110.1139242984811), $MachinePrecision]}, If[Less[x, -3.326128725870005e+62], t$95$0, If[Less[x, 9.429991714554673e+55], N[(N[(N[(x - 2.0), $MachinePrecision] / 1.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(N[(x * 4.16438922228), $MachinePrecision] + 78.6994924154), $MachinePrecision] * x), $MachinePrecision] + 137.519416416), $MachinePrecision] * x), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision] + z), $MachinePrecision] / N[(N[(N[(N[(N[(263.505074721 * x), $MachinePrecision] + N[(N[(43.3400022514 * N[(x * x), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision] * x), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -332612872587000500000000000000000000000000000000000000000000000) (- (+ (/ y (* x x)) (* 104109730557/25000000000 x)) 1101139242984811/10000000000000) (if (< x 94299917145546730000000000000000000000000000000000000000) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* x 104109730557/25000000000) 393497462077/5000000000) x) 4297481763/31250000) x) y) x) z) (+ (* (+ (+ (* 263505074721/1000000000 x) (+ (* 216700011257/5000000000 (* x x)) (* x (* x x)))) 156699607947/500000000) x) 23533438303/500000000))) (- (+ (/ y (* x x)) (* 104109730557/25000000000 x)) 1101139242984811/10000000000000))))

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