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

Percentage Accurate: 57.9% → 98.5%

Time: 13.5s

Alternatives: 16

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: 57.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \leq 10^{+304}:\\ \;\;\;\;\frac{\left(x \cdot x - 4\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}}{2 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{\frac{\frac{130977.50649958357 - y}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x} - 4.16438922228\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          (+
           (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x)
           y)
          x)
         z))
       (+
        (*
         (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894)
         x)
        47.066876606))
      1e+304)
   (/
    (*
     (- (* x x) 4.0)
     (/
      (fma
       (fma (fma (fma 4.16438922228 x 78.6994924154) x 137.519416416) x y)
       x
       z)
      (fma
       (fma (fma (+ 43.3400022514 x) x 263.505074721) x 313.399215894)
       x
       47.066876606)))
    (+ 2.0 x))
   (*
    (- x)
    (-
     (/
      (+
       (/ (- (/ (- 130977.50649958357 y) x) 3655.1204654076414) x)
       110.1139242984811)
      x)
     4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606)) <= 1e+304) {
		tmp = (((x * x) - 4.0) * (fma(fma(fma(fma(4.16438922228, x, 78.6994924154), x, 137.519416416), x, y), x, z) / fma(fma(fma((43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606))) / (2.0 + x);
	} else {
		tmp = -x * (((((((130977.50649958357 - y) / x) - 3655.1204654076414) / x) + 110.1139242984811) / x) - 4.16438922228);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(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)) <= 1e+304)
		tmp = Float64(Float64(Float64(Float64(x * x) - 4.0) * Float64(fma(fma(fma(fma(4.16438922228, x, 78.6994924154), x, 137.519416416), x, y), x, z) / fma(fma(fma(Float64(43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606))) / Float64(2.0 + x));
	else
		tmp = Float64(Float64(-x) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(130977.50649958357 - y) / x) - 3655.1204654076414) / x) + 110.1139242984811) / x) - 4.16438922228));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(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], 1e+304], N[(N[(N[(N[(x * x), $MachinePrecision] - 4.0), $MachinePrecision] * N[(N[(N[(N[(N[(4.16438922228 * x + 78.6994924154), $MachinePrecision] * x + 137.519416416), $MachinePrecision] * x + y), $MachinePrecision] * x + z), $MachinePrecision] / N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x + 263.505074721), $MachinePrecision] * x + 313.399215894), $MachinePrecision] * x + 47.066876606), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + x), $MachinePrecision]), $MachinePrecision], N[((-x) * N[(N[(N[(N[(N[(N[(N[(130977.50649958357 - y), $MachinePrecision] / x), $MachinePrecision] - 3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] - 4.16438922228), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64))) < 9.9999999999999994e303

   1. Initial program 98.9%

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

   4. Applied rewrites 99.5%

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

   if 9.9999999999999994e303 < (/.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 0.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   8. Applied rewrites 97.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \leq 10^{+304}:\\ \;\;\;\;\frac{\left(x \cdot x - 4\right) \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}}{2 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{\frac{\frac{130977.50649958357 - y}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x} - 4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \leq 10^{+304}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \left(x - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{\frac{\frac{130977.50649958357 - y}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x} - 4.16438922228\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (- x 2.0)
        (+
         (*
          (+
           (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x)
           y)
          x)
         z))
       (+
        (*
         (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894)
         x)
        47.066876606))
      1e+304)
   (*
    (/
     (fma
      (fma (fma (fma 4.16438922228 x 78.6994924154) x 137.519416416) x y)
      x
      z)
     (fma
      (fma (fma (+ 43.3400022514 x) x 263.505074721) x 313.399215894)
      x
      47.066876606))
    (- x 2.0))
   (*
    (- x)
    (-
     (/
      (+
       (/ (- (/ (- 130977.50649958357 y) x) 3655.1204654076414) x)
       110.1139242984811)
      x)
     4.16438922228))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((x - 2.0) * ((((((((x * 4.16438922228) + 78.6994924154) * x) + 137.519416416) * x) + y) * x) + z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606)) <= 1e+304) {
		tmp = (fma(fma(fma(fma(4.16438922228, x, 78.6994924154), x, 137.519416416), x, y), x, z) / fma(fma(fma((43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606)) * (x - 2.0);
	} else {
		tmp = -x * (((((((130977.50649958357 - y) / x) - 3655.1204654076414) / x) + 110.1139242984811) / x) - 4.16438922228);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(x - 2.0) * Float64(Float64(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)) <= 1e+304)
		tmp = Float64(Float64(fma(fma(fma(fma(4.16438922228, x, 78.6994924154), x, 137.519416416), x, y), x, z) / fma(fma(fma(Float64(43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606)) * Float64(x - 2.0));
	else
		tmp = Float64(Float64(-x) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(130977.50649958357 - y) / x) - 3655.1204654076414) / x) + 110.1139242984811) / x) - 4.16438922228));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(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], 1e+304], N[(N[(N[(N[(N[(N[(4.16438922228 * x + 78.6994924154), $MachinePrecision] * x + 137.519416416), $MachinePrecision] * x + y), $MachinePrecision] * x + z), $MachinePrecision] / N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x + 263.505074721), $MachinePrecision] * x + 313.399215894), $MachinePrecision] * x + 47.066876606), $MachinePrecision]), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision], N[((-x) * N[(N[(N[(N[(N[(N[(N[(130977.50649958357 - y), $MachinePrecision] / x), $MachinePrecision] - 3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] - 4.16438922228), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64))) < 9.9999999999999994e303

   1. Initial program 98.9%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   2. Add Preprocessing
   3. 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 99.5%

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

   if 9.9999999999999994e303 < (/.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 0.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   8. Applied rewrites 97.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \leq 10^{+304}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \left(x - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{\frac{\frac{130977.50649958357 - y}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x} - 4.16438922228\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1750000 \lor \neg \left(x \leq 1.7 \cdot 10^{+14}\right):\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{\frac{\frac{130977.50649958357 - y}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x} - 4.16438922228\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(78.6994924154, x, 137.519416416\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1750000.0) (not (<= x 1.7e+14)))
   (*
    (- x)
    (-
     (/
      (+
       (/ (- (/ (- 130977.50649958357 y) x) 3655.1204654076414) x)
       110.1139242984811)
      x)
     4.16438922228))
   (/
    (* (- x 2.0) (fma (fma (fma 78.6994924154 x 137.519416416) x y) x z))
    (fma
     (fma (fma (+ 43.3400022514 x) x 263.505074721) x 313.399215894)
     x
     47.066876606))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1750000.0) || !(x <= 1.7e+14)) {
		tmp = -x * (((((((130977.50649958357 - y) / x) - 3655.1204654076414) / x) + 110.1139242984811) / x) - 4.16438922228);
	} else {
		tmp = ((x - 2.0) * fma(fma(fma(78.6994924154, x, 137.519416416), x, y), x, z)) / fma(fma(fma((43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1750000.0) || !(x <= 1.7e+14))
		tmp = Float64(Float64(-x) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(130977.50649958357 - y) / x) - 3655.1204654076414) / x) + 110.1139242984811) / x) - 4.16438922228));
	else
		tmp = Float64(Float64(Float64(x - 2.0) * fma(fma(fma(78.6994924154, x, 137.519416416), x, y), x, z)) / fma(fma(fma(Float64(43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1750000.0], N[Not[LessEqual[x, 1.7e+14]], $MachinePrecision]], N[((-x) * N[(N[(N[(N[(N[(N[(N[(130977.50649958357 - y), $MachinePrecision] / x), $MachinePrecision] - 3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] - 4.16438922228), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(N[(78.6994924154 * x + 137.519416416), $MachinePrecision] * x + y), $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x + 263.505074721), $MachinePrecision] * x + 313.399215894), $MachinePrecision] * x + 47.066876606), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e6 or 1.7e14 < 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

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

      1. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   8. Applied rewrites 95.5%

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

   if -1.75e6 < x < 1.7e14

   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%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 99.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1750000 \lor \neg \left(x \leq 1.7 \cdot 10^{+14}\right):\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{\frac{\frac{130977.50649958357 - y}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x} - 4.16438922228\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(78.6994924154, x, 137.519416416\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1750000 \lor \neg \left(x \leq 1.7 \cdot 10^{+14}\right):\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{\frac{\frac{130977.50649958357 - y}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x} - 4.16438922228\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1750000.0) (not (<= x 1.7e+14)))
   (*
    (- x)
    (-
     (/
      (+
       (/ (- (/ (- 130977.50649958357 y) x) 3655.1204654076414) x)
       110.1139242984811)
      x)
     4.16438922228))
   (/
    (* (- x 2.0) (fma (fma 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) {
	double tmp;
	if ((x <= -1750000.0) || !(x <= 1.7e+14)) {
		tmp = -x * (((((((130977.50649958357 - y) / x) - 3655.1204654076414) / x) + 110.1139242984811) / x) - 4.16438922228);
	} else {
		tmp = ((x - 2.0) * fma(fma(137.519416416, x, y), x, z)) / (((((((x + 43.3400022514) * x) + 263.505074721) * x) + 313.399215894) * x) + 47.066876606);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1750000.0) || !(x <= 1.7e+14))
		tmp = Float64(Float64(-x) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(130977.50649958357 - y) / x) - 3655.1204654076414) / x) + 110.1139242984811) / x) - 4.16438922228));
	else
		tmp = Float64(Float64(Float64(x - 2.0) * fma(fma(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
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1750000.0], N[Not[LessEqual[x, 1.7e+14]], $MachinePrecision]], N[((-x) * N[(N[(N[(N[(N[(N[(N[(130977.50649958357 - y), $MachinePrecision] / x), $MachinePrecision] - 3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] - 4.16438922228), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(137.519416416 * x + y), $MachinePrecision] * x + 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]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e6 or 1.7e14 < 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

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

      1. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   8. Applied rewrites 95.5%

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

   if -1.75e6 < x < 1.7e14

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1750000 \lor \neg \left(x \leq 1.7 \cdot 10^{+14}\right):\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{\frac{\frac{130977.50649958357 - y}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x} - 4.16438922228\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1750000 \lor \neg \left(x \leq 1.7 \cdot 10^{+14}\right):\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{\frac{\frac{130977.50649958357 - y}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x} - 4.16438922228\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \left(x - 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1750000.0) (not (<= x 1.7e+14)))
   (*
    (- x)
    (-
     (/
      (+
       (/ (- (/ (- 130977.50649958357 y) x) 3655.1204654076414) x)
       110.1139242984811)
      x)
     4.16438922228))
   (*
    (/
     (fma (fma 137.519416416 x y) x z)
     (fma
      (fma (fma (+ 43.3400022514 x) x 263.505074721) x 313.399215894)
      x
      47.066876606))
    (- x 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1750000.0) || !(x <= 1.7e+14)) {
		tmp = -x * (((((((130977.50649958357 - y) / x) - 3655.1204654076414) / x) + 110.1139242984811) / x) - 4.16438922228);
	} else {
		tmp = (fma(fma(137.519416416, x, y), x, z) / fma(fma(fma((43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606)) * (x - 2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1750000.0) || !(x <= 1.7e+14))
		tmp = Float64(Float64(-x) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(130977.50649958357 - y) / x) - 3655.1204654076414) / x) + 110.1139242984811) / x) - 4.16438922228));
	else
		tmp = Float64(Float64(fma(fma(137.519416416, x, y), x, z) / fma(fma(fma(Float64(43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606)) * Float64(x - 2.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1750000.0], N[Not[LessEqual[x, 1.7e+14]], $MachinePrecision]], N[((-x) * N[(N[(N[(N[(N[(N[(N[(130977.50649958357 - y), $MachinePrecision] / x), $MachinePrecision] - 3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] - 4.16438922228), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(137.519416416 * x + y), $MachinePrecision] * x + z), $MachinePrecision] / N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x + 263.505074721), $MachinePrecision] * x + 313.399215894), $MachinePrecision] * x + 47.066876606), $MachinePrecision]), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75e6 or 1.7e14 < 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

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

      1. lower-*.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in x around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   8. Applied rewrites 95.5%

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

   if -1.75e6 < x < 1.7e14

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 98.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1750000 \lor \neg \left(x \leq 1.7 \cdot 10^{+14}\right):\\ \;\;\;\;\left(-x\right) \cdot \left(\frac{\frac{\frac{130977.50649958357 - y}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x} - 4.16438922228\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \left(x - 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -115000000:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \left(x - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -115000000.0)
   (* (+ (/ (- (/ 3655.1204654076414 x) 110.1139242984811) x) 4.16438922228) x)
   (if (<= x 2.9e+14)
     (*
      (/
       (fma (fma 137.519416416 x y) x z)
       (fma
        (fma (fma (+ 43.3400022514 x) x 263.505074721) x 313.399215894)
        x
        47.066876606))
      (- x 2.0))
     (* (- 4.16438922228 (/ 110.1139242984811 x)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -115000000.0) {
		tmp = ((((3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x;
	} else if (x <= 2.9e+14) {
		tmp = (fma(fma(137.519416416, x, y), x, z) / fma(fma(fma((43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606)) * (x - 2.0);
	} else {
		tmp = (4.16438922228 - (110.1139242984811 / x)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -115000000.0)
		tmp = Float64(Float64(Float64(Float64(Float64(3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x);
	elseif (x <= 2.9e+14)
		tmp = Float64(Float64(fma(fma(137.519416416, x, y), x, z) / fma(fma(fma(Float64(43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606)) * Float64(x - 2.0));
	else
		tmp = Float64(Float64(4.16438922228 - Float64(110.1139242984811 / x)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -115000000.0], N[(N[(N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] + 4.16438922228), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 2.9e+14], N[(N[(N[(N[(137.519416416 * x + y), $MachinePrecision] * x + z), $MachinePrecision] / N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x + 263.505074721), $MachinePrecision] * x + 313.399215894), $MachinePrecision] * x + 47.066876606), $MachinePrecision]), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(4.16438922228 - N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.15e8

   1. Initial program 13.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 90.1

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

   5. Applied rewrites 90.1%

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

   if -1.15e8 < x < 2.9e14

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.2

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

   5. Applied rewrites 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 98.2%

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

   if 2.9e14 < x

   1. Initial program 16.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

Alternative 7: 92.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -38:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(78.6994924154, x, 137.519416416\right), x, y\right) \cdot x + z\right)}{\mathsf{fma}\left(313.399215894, x, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -38.0)
   (* (+ (/ (- (/ 3655.1204654076414 x) 110.1139242984811) x) 4.16438922228) x)
   (if (<= x 1.15e+14)
     (/
      (* (- x 2.0) (+ (* (fma (fma 78.6994924154 x 137.519416416) x y) x) z))
      (fma 313.399215894 x 47.066876606))
     (* (- 4.16438922228 (/ 110.1139242984811 x)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -38.0) {
		tmp = ((((3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x;
	} else if (x <= 1.15e+14) {
		tmp = ((x - 2.0) * ((fma(fma(78.6994924154, x, 137.519416416), x, y) * x) + z)) / fma(313.399215894, x, 47.066876606);
	} else {
		tmp = (4.16438922228 - (110.1139242984811 / x)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -38.0)
		tmp = Float64(Float64(Float64(Float64(Float64(3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x);
	elseif (x <= 1.15e+14)
		tmp = Float64(Float64(Float64(x - 2.0) * Float64(Float64(fma(fma(78.6994924154, x, 137.519416416), x, y) * x) + z)) / fma(313.399215894, x, 47.066876606));
	else
		tmp = Float64(Float64(4.16438922228 - Float64(110.1139242984811 / x)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -38.0], N[(N[(N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] + 4.16438922228), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.15e+14], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(N[(N[(78.6994924154 * x + 137.519416416), $MachinePrecision] * x + y), $MachinePrecision] * x), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(313.399215894 * x + 47.066876606), $MachinePrecision]), $MachinePrecision], N[(N[(4.16438922228 - N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -38

   1. Initial program 15.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if -38 < x < 1.15e14

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.6

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.0

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

   7. Applied rewrites 97.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 97.0

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

   10. Applied rewrites 97.0%

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

   if 1.15e14 < x

   1. Initial program 16.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

Alternative 8: 92.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -38:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(137.519416416 \cdot x + y\right) \cdot x + z\right)}{\mathsf{fma}\left(313.399215894, x, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -38.0)
   (* (+ (/ (- (/ 3655.1204654076414 x) 110.1139242984811) x) 4.16438922228) x)
   (if (<= x 1.15e+14)
     (/
      (* (- x 2.0) (+ (* (+ (* 137.519416416 x) y) x) z))
      (fma 313.399215894 x 47.066876606))
     (* (- 4.16438922228 (/ 110.1139242984811 x)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -38.0) {
		tmp = ((((3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x;
	} else if (x <= 1.15e+14) {
		tmp = ((x - 2.0) * ((((137.519416416 * x) + y) * x) + z)) / fma(313.399215894, x, 47.066876606);
	} else {
		tmp = (4.16438922228 - (110.1139242984811 / x)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -38.0)
		tmp = Float64(Float64(Float64(Float64(Float64(3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x);
	elseif (x <= 1.15e+14)
		tmp = Float64(Float64(Float64(x - 2.0) * Float64(Float64(Float64(Float64(137.519416416 * x) + y) * x) + z)) / fma(313.399215894, x, 47.066876606));
	else
		tmp = Float64(Float64(4.16438922228 - Float64(110.1139242984811 / x)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -38.0], N[(N[(N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] + 4.16438922228), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.15e+14], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(N[(N[(137.519416416 * x), $MachinePrecision] + y), $MachinePrecision] * x), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] / N[(313.399215894 * x + 47.066876606), $MachinePrecision]), $MachinePrecision], N[(N[(4.16438922228 - N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -38

   1. Initial program 15.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if -38 < x < 1.15e14

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.6

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.0

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

   7. Applied rewrites 97.0%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 96.7

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

   10. Applied rewrites 96.7%

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

   if 1.15e14 < x

   1. Initial program 16.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

Alternative 9: 92.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -38:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right)}{\mathsf{fma}\left(313.399215894, x, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -38.0)
   (* (+ (/ (- (/ 3655.1204654076414 x) 110.1139242984811) x) 4.16438922228) x)
   (if (<= x 1.15e+14)
     (/
      (* (- x 2.0) (fma (fma 137.519416416 x y) x z))
      (fma 313.399215894 x 47.066876606))
     (* (- 4.16438922228 (/ 110.1139242984811 x)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -38.0) {
		tmp = ((((3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x;
	} else if (x <= 1.15e+14) {
		tmp = ((x - 2.0) * fma(fma(137.519416416, x, y), x, z)) / fma(313.399215894, x, 47.066876606);
	} else {
		tmp = (4.16438922228 - (110.1139242984811 / x)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -38.0)
		tmp = Float64(Float64(Float64(Float64(Float64(3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x);
	elseif (x <= 1.15e+14)
		tmp = Float64(Float64(Float64(x - 2.0) * fma(fma(137.519416416, x, y), x, z)) / fma(313.399215894, x, 47.066876606));
	else
		tmp = Float64(Float64(4.16438922228 - Float64(110.1139242984811 / x)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -38.0], N[(N[(N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] + 4.16438922228), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.15e+14], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(137.519416416 * x + y), $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision] / N[(313.399215894 * x + 47.066876606), $MachinePrecision]), $MachinePrecision], N[(N[(4.16438922228 - N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -38

   1. Initial program 15.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if -38 < x < 1.15e14

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 96.7

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

   8. Applied rewrites 96.7%

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

   if 1.15e14 < x

   1. Initial program 16.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

Alternative 10: 91.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -280 \lor \neg \left(x \leq 1.15 \cdot 10^{+14}\right):\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right)}{47.066876606}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -280.0) (not (<= x 1.15e+14)))
   (* (- 4.16438922228 (/ 110.1139242984811 x)) x)
   (/ (* (- x 2.0) (fma (fma 137.519416416 x y) x z)) 47.066876606)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -280.0) || !(x <= 1.15e+14)) {
		tmp = (4.16438922228 - (110.1139242984811 / x)) * x;
	} else {
		tmp = ((x - 2.0) * fma(fma(137.519416416, x, y), x, z)) / 47.066876606;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -280.0) || !(x <= 1.15e+14))
		tmp = Float64(Float64(4.16438922228 - Float64(110.1139242984811 / x)) * x);
	else
		tmp = Float64(Float64(Float64(x - 2.0) * fma(fma(137.519416416, x, y), x, z)) / 47.066876606);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -280.0], N[Not[LessEqual[x, 1.15e+14]], $MachinePrecision]], N[(N[(4.16438922228 - N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(137.519416416 * x + y), $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision] / 47.066876606), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -280 or 1.15e14 < 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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 91.1

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

   5. Applied rewrites 91.1%

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

   if -280 < x < 1.15e14

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.5%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -280 \lor \neg \left(x \leq 1.15 \cdot 10^{+14}\right):\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right)}{47.066876606}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -260:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+14}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right)}{47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -260.0)
   (* (+ (/ (- (/ 3655.1204654076414 x) 110.1139242984811) x) 4.16438922228) x)
   (if (<= x 1.15e+14)
     (/ (* (- x 2.0) (fma (fma 137.519416416 x y) x z)) 47.066876606)
     (* (- 4.16438922228 (/ 110.1139242984811 x)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -260.0) {
		tmp = ((((3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x;
	} else if (x <= 1.15e+14) {
		tmp = ((x - 2.0) * fma(fma(137.519416416, x, y), x, z)) / 47.066876606;
	} else {
		tmp = (4.16438922228 - (110.1139242984811 / x)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -260.0)
		tmp = Float64(Float64(Float64(Float64(Float64(3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x);
	elseif (x <= 1.15e+14)
		tmp = Float64(Float64(Float64(x - 2.0) * fma(fma(137.519416416, x, y), x, z)) / 47.066876606);
	else
		tmp = Float64(Float64(4.16438922228 - Float64(110.1139242984811 / x)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -260.0], N[(N[(N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] + 4.16438922228), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.15e+14], N[(N[(N[(x - 2.0), $MachinePrecision] * N[(N[(137.519416416 * x + y), $MachinePrecision] * x + z), $MachinePrecision]), $MachinePrecision] / 47.066876606), $MachinePrecision], N[(N[(4.16438922228 - N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -260

   1. Initial program 15.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if -260 < x < 1.15e14

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.5%

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

   if 1.15e14 < x

   1. Initial program 16.8%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 95.1

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

   5. Applied rewrites 95.1%

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

Alternative 12: 89.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.23 \lor \neg \left(x \leq 1.45 \cdot 10^{-9}\right):\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, 0.3041881842569256 \cdot z\right), x, -0.0424927283095952 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.23) (not (<= x 1.45e-9)))
   (* (- 4.16438922228 (/ 110.1139242984811 x)) x)
   (fma
    (fma -0.0424927283095952 y (* 0.3041881842569256 z))
    x
    (* -0.0424927283095952 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -0.23) || !(x <= 1.45e-9)) {
		tmp = (4.16438922228 - (110.1139242984811 / x)) * x;
	} else {
		tmp = fma(fma(-0.0424927283095952, y, (0.3041881842569256 * z)), x, (-0.0424927283095952 * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -0.23) || !(x <= 1.45e-9))
		tmp = Float64(Float64(4.16438922228 - Float64(110.1139242984811 / x)) * x);
	else
		tmp = fma(fma(-0.0424927283095952, y, Float64(0.3041881842569256 * z)), x, Float64(-0.0424927283095952 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.23], N[Not[LessEqual[x, 1.45e-9]], $MachinePrecision]], N[(N[(4.16438922228 - N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(-0.0424927283095952 * y + N[(0.3041881842569256 * z), $MachinePrecision]), $MachinePrecision] * x + N[(-0.0424927283095952 * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.23000000000000001 or 1.44999999999999996e-9 < x

   1. Initial program 18.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -0.23000000000000001 < x < 1.44999999999999996e-9

   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%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 91.4

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

   7. Applied rewrites 91.4%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.23 \lor \neg \left(x \leq 1.45 \cdot 10^{-9}\right):\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0424927283095952, y, 0.3041881842569256 \cdot z\right), x, -0.0424927283095952 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.1% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -0.175) (not (<= x 2.0)))
   (* (- 4.16438922228 (/ 110.1139242984811 x)) x)
   (fma (* -5.843575199059173 x) x (* -0.0424927283095952 z))))

C

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

Julia

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

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -0.175], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(N[(4.16438922228 - N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(-5.843575199059173 * x), $MachinePrecision] * x + N[(-0.0424927283095952 * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.17499999999999999 or 2 < x

   1. Initial program 17.4%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if -0.17499999999999999 < x < 2

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{-1000000000}{23533438303} \cdot z + \color{blue}{x \cdot \left(\left(-1 \cdot \left(x \cdot \left(\frac{137519416416}{23533438303} + \left(\frac{-131752537360500000000}{553822718361107519809} \cdot z + \frac{156699607947}{23533438303} \cdot \left(\frac{500000000}{23533438303} \cdot z - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right)\right)\right) + \frac{500000000}{23533438303} \cdot z\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-x, \mathsf{fma}\left(z \cdot 0.3041881842569256, 6.658593866711955, -0.23789659216289816 \cdot z\right) + 5.843575199059173, z \cdot 0.3041881842569256\right), \color{blue}{x}, -0.0424927283095952 \cdot z\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{-137519416416}{23533438303} \cdot x, x, \frac{-1000000000}{23533438303} \cdot z\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 71.0%

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

4. Final simplification 79.9%

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

Alternative 14: 79.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -350 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;4.16438922228 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-5.843575199059173 \cdot x, x, -0.0424927283095952 \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -350.0) (not (<= x 2.0)))
   (* 4.16438922228 x)
   (fma (* -5.843575199059173 x) x (* -0.0424927283095952 z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -350.0) || !(x <= 2.0)) {
		tmp = 4.16438922228 * x;
	} else {
		tmp = fma((-5.843575199059173 * x), x, (-0.0424927283095952 * z));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -350.0) || !(x <= 2.0))
		tmp = Float64(4.16438922228 * x);
	else
		tmp = fma(Float64(-5.843575199059173 * x), x, Float64(-0.0424927283095952 * z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -350.0], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(4.16438922228 * x), $MachinePrecision], N[(N[(-5.843575199059173 * x), $MachinePrecision] * x + N[(-0.0424927283095952 * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -350 or 2 < x

   1. Initial program 17.4%

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

   4. Applied rewrites 22.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

   if -350 < x < 2

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \frac{-1000000000}{23533438303} \cdot z + \color{blue}{x \cdot \left(\left(-1 \cdot \left(x \cdot \left(\frac{137519416416}{23533438303} + \left(\frac{-131752537360500000000}{553822718361107519809} \cdot z + \frac{156699607947}{23533438303} \cdot \left(\frac{500000000}{23533438303} \cdot z - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)\right)\right)\right) + \frac{500000000}{23533438303} \cdot z\right) - \frac{-156699607947000000000}{553822718361107519809} \cdot z\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-x, \mathsf{fma}\left(z \cdot 0.3041881842569256, 6.658593866711955, -0.23789659216289816 \cdot z\right) + 5.843575199059173, z \cdot 0.3041881842569256\right), \color{blue}{x}, -0.0424927283095952 \cdot z\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{-137519416416}{23533438303} \cdot x, x, \frac{-1000000000}{23533438303} \cdot z\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 71.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -350 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;4.16438922228 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-5.843575199059173 \cdot x, x, -0.0424927283095952 \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;4.16438922228 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.0424927283095952 \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.5) (not (<= x 2.0)))
   (* 4.16438922228 x)
   (* -0.0424927283095952 z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.5) || !(x <= 2.0)) {
		tmp = 4.16438922228 * x;
	} else {
		tmp = -0.0424927283095952 * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-5.5d0)) .or. (.not. (x <= 2.0d0))) then
        tmp = 4.16438922228d0 * x
    else
        tmp = (-0.0424927283095952d0) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.5) || !(x <= 2.0)) {
		tmp = 4.16438922228 * x;
	} else {
		tmp = -0.0424927283095952 * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.5) or not (x <= 2.0):
		tmp = 4.16438922228 * x
	else:
		tmp = -0.0424927283095952 * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.5) || !(x <= 2.0))
		tmp = Float64(4.16438922228 * x);
	else
		tmp = Float64(-0.0424927283095952 * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.5) || ~((x <= 2.0)))
		tmp = 4.16438922228 * x;
	else
		tmp = -0.0424927283095952 * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.5], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(4.16438922228 * x), $MachinePrecision], N[(-0.0424927283095952 * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5 or 2 < x

   1. Initial program 17.4%

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

   4. Applied rewrites 22.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

   if -5.5 < x < 2

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 65.0

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

   5. Applied rewrites 65.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;4.16438922228 \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.0424927283095952 \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 34.5% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ -0.0424927283095952 \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* -0.0424927283095952 z))

C

double code(double x, double y, double z) {
	return -0.0424927283095952 * z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (-0.0424927283095952d0) * z
end function

Java

public static double code(double x, double y, double z) {
	return -0.0424927283095952 * z;
}

Python

def code(x, y, z):
	return -0.0424927283095952 * z

Julia

function code(x, y, z)
	return Float64(-0.0424927283095952 * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -0.0424927283095952 * z;
end

Wolfram

code[x_, y_, z_] := N[(-0.0424927283095952 * z), $MachinePrecision]

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 35.5

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

5. Applied rewrites 35.5%

   \[\leadsto \color{blue}{-0.0424927283095952 \cdot z} \]
6. Add Preprocessing

Developer Target 1: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024329 
(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)))