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

Percentage Accurate: 57.5% → 96.9%

Time: 14.6s

Alternatives: 17

Speedup: 4.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 57.5% 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: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 14200000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(\left(43.3400022514 + x\right) \cdot x + 263.505074721\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          (-
           (/
            (-
             -110.1139242984811
             (/
              (- (+ (/ 130977.50649958357 x) -3655.1204654076414) (/ y x))
              x))
            x)
           -4.16438922228)
          x)))
   (if (<= x -4800000.0)
     t_0
     (if (<= x 14200000000000.0)
       (/
        (* (fma (fma 137.519416416 x y) x z) (- x 2.0))
        (+
         47.066876606
         (*
          (+ 313.399215894 (* (+ (* (+ 43.3400022514 x) x) 263.505074721) x))
          x)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((-110.1139242984811 - ((((130977.50649958357 / x) + -3655.1204654076414) - (y / x)) / x)) / x) - -4.16438922228) * x;
	double tmp;
	if (x <= -4800000.0) {
		tmp = t_0;
	} else if (x <= 14200000000000.0) {
		tmp = (fma(fma(137.519416416, x, y), x, z) * (x - 2.0)) / (47.066876606 + ((313.399215894 + ((((43.3400022514 + x) * x) + 263.505074721) * x)) * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(-110.1139242984811 - Float64(Float64(Float64(Float64(130977.50649958357 / x) + -3655.1204654076414) - Float64(y / x)) / x)) / x) - -4.16438922228) * x)
	tmp = 0.0
	if (x <= -4800000.0)
		tmp = t_0;
	elseif (x <= 14200000000000.0)
		tmp = Float64(Float64(fma(fma(137.519416416, x, y), x, z) * Float64(x - 2.0)) / Float64(47.066876606 + Float64(Float64(313.399215894 + Float64(Float64(Float64(Float64(43.3400022514 + x) * x) + 263.505074721) * x)) * x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(-110.1139242984811 - N[(N[(N[(N[(130977.50649958357 / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4800000.0], t$95$0, If[LessEqual[x, 14200000000000.0], N[(N[(N[(N[(137.519416416 * x + y), $MachinePrecision] * x + z), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(47.066876606 + N[(N[(313.399215894 + N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x), $MachinePrecision] + 263.505074721), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.8e6 or 1.42e13 < x

   1. Initial program 15.4%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]
   4. 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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 95.9%

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

   if -4.8e6 < x < 1.42e13

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \mathbf{elif}\;x \leq 14200000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(\left(43.3400022514 + x\right) \cdot x + 263.505074721\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(z + \left(y + \left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(\left(43.3400022514 + x\right) \cdot x + 263.505074721\right) \cdot x\right) \cdot x} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \mathsf{fma}\left(x \cdot x, x, -8\right)}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(2, x, 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (+
         z
         (*
          (+
           y
           (* (+ 137.519416416 (* (+ 78.6994924154 (* 4.16438922228 x)) x)) x))
          x))
        (- x 2.0))
       (+
        47.066876606
        (*
         (+ 313.399215894 (* (+ (* (+ 43.3400022514 x) x) 263.505074721) x))
         x)))
      INFINITY)
   (/
    (*
     (/
      (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))
     (fma (* x x) x -8.0))
    (fma x x (fma 2.0 x 4.0)))
   (*
    (-
     (/
      (-
       -110.1139242984811
       (/ (- (+ (/ 130977.50649958357 x) -3655.1204654076414) (/ y x)) x))
      x)
     -4.16438922228)
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((z + ((y + ((137.519416416 + ((78.6994924154 + (4.16438922228 * x)) * x)) * x)) * x)) * (x - 2.0)) / (47.066876606 + ((313.399215894 + ((((43.3400022514 + x) * x) + 263.505074721) * x)) * x))) <= ((double) INFINITY)) {
		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)) * fma((x * x), x, -8.0)) / fma(x, x, fma(2.0, x, 4.0));
	} else {
		tmp = (((-110.1139242984811 - ((((130977.50649958357 / x) + -3655.1204654076414) - (y / x)) / x)) / x) - -4.16438922228) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(z + Float64(Float64(y + Float64(Float64(137.519416416 + Float64(Float64(78.6994924154 + Float64(4.16438922228 * x)) * x)) * x)) * x)) * Float64(x - 2.0)) / Float64(47.066876606 + Float64(Float64(313.399215894 + Float64(Float64(Float64(Float64(43.3400022514 + x) * x) + 263.505074721) * x)) * x))) <= Inf)
		tmp = Float64(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)) * fma(Float64(x * x), x, -8.0)) / fma(x, x, fma(2.0, x, 4.0)));
	else
		tmp = Float64(Float64(Float64(Float64(-110.1139242984811 - Float64(Float64(Float64(Float64(130977.50649958357 / x) + -3655.1204654076414) - Float64(y / x)) / x)) / x) - -4.16438922228) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(z + N[(N[(y + N[(N[(137.519416416 + N[(N[(78.6994924154 + N[(4.16438922228 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(47.066876606 + N[(N[(313.399215894 + N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x), $MachinePrecision] + 263.505074721), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(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[(N[(x * x), $MachinePrecision] * x + -8.0), $MachinePrecision]), $MachinePrecision] / N[(x * x + N[(2.0 * x + 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(-110.1139242984811 - N[(N[(N[(N[(130977.50649958357 / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.7%

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

   4. Applied rewrites 96.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]
   4. 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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 99.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z + \left(y + \left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(\left(43.3400022514 + x\right) \cdot x + 263.505074721\right) \cdot x\right) \cdot x} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \mathsf{fma}\left(x \cdot x, x, -8\right)}{\mathsf{fma}\left(x, x, \mathsf{fma}\left(2, x, 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(z + \left(y + \left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(\left(43.3400022514 + x\right) \cdot x + 263.505074721\right) \cdot x\right) \cdot x} \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \left(x - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (+
         z
         (*
          (+
           y
           (* (+ 137.519416416 (* (+ 78.6994924154 (* 4.16438922228 x)) x)) x))
          x))
        (- x 2.0))
       (+
        47.066876606
        (*
         (+ 313.399215894 (* (+ (* (+ 43.3400022514 x) x) 263.505074721) x))
         x)))
      5e+286)
   (*
    (/
     (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))
   (*
    (-
     (/
      (-
       -110.1139242984811
       (/ (- (+ (/ 130977.50649958357 x) -3655.1204654076414) (/ y x)) x))
      x)
     -4.16438922228)
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((z + ((y + ((137.519416416 + ((78.6994924154 + (4.16438922228 * x)) * x)) * x)) * x)) * (x - 2.0)) / (47.066876606 + ((313.399215894 + ((((43.3400022514 + x) * x) + 263.505074721) * x)) * x))) <= 5e+286) {
		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 = (((-110.1139242984811 - ((((130977.50649958357 / x) + -3655.1204654076414) - (y / x)) / x)) / x) - -4.16438922228) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(z + Float64(Float64(y + Float64(Float64(137.519416416 + Float64(Float64(78.6994924154 + Float64(4.16438922228 * x)) * x)) * x)) * x)) * Float64(x - 2.0)) / Float64(47.066876606 + Float64(Float64(313.399215894 + Float64(Float64(Float64(Float64(43.3400022514 + x) * x) + 263.505074721) * x)) * x))) <= 5e+286)
		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(Float64(Float64(-110.1139242984811 - Float64(Float64(Float64(Float64(130977.50649958357 / x) + -3655.1204654076414) - Float64(y / x)) / x)) / x) - -4.16438922228) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(z + N[(N[(y + N[(N[(137.519416416 + N[(N[(78.6994924154 + N[(4.16438922228 * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(47.066876606 + N[(N[(313.399215894 + N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x), $MachinePrecision] + 263.505074721), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+286], 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[(N[(N[(N[(-110.1139242984811 - N[(N[(N[(N[(130977.50649958357 / x), $MachinePrecision] + -3655.1204654076414), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - -4.16438922228), $MachinePrecision] * x), $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))) < 5.0000000000000004e286

   1. Initial program 95.8%

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

      1. 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 97.0%

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

   if 5.0000000000000004e286 < (/.f64 (*.f64 (-.f64 x #s(literal 2 binary64)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x #s(literal 104109730557/25000000000 binary64)) #s(literal 393497462077/5000000000 binary64)) x) #s(literal 4297481763/31250000 binary64)) x) y) x) z)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 x #s(literal 216700011257/5000000000 binary64)) x) #s(literal 263505074721/1000000000 binary64)) x) #s(literal 156699607947/500000000 binary64)) x) #s(literal 23533438303/500000000 binary64)))

   1. Initial program 0.1%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{-1 \cdot \frac{\left(-1 \cdot \frac{y}{x} + \frac{409304707811198655637810418659684985388407301}{3125000000000000000000000000000000000000} \cdot \frac{1}{x}\right) - \frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{x} - \frac{13764240537310136880149}{125000000000000000000}}{x} - \frac{104109730557}{25000000000}\right)\right)} \]
   4. 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. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 98.2%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z + \left(y + \left(137.519416416 + \left(78.6994924154 + 4.16438922228 \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(\left(43.3400022514 + x\right) \cdot x + 263.505074721\right) \cdot x\right) \cdot x} \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.16438922228, x, 78.6994924154\right), x, 137.519416416\right), x, y\right), x, z\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \left(x - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-110.1139242984811 - \frac{\left(\frac{130977.50649958357}{x} + -3655.1204654076414\right) - \frac{y}{x}}{x}}{x} - -4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;\left(4.16438922228 - \frac{\frac{\frac{130977.50649958357}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x}\right) \cdot x\\ \mathbf{elif}\;x \leq 14200000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(\left(43.3400022514 + x\right) \cdot x + 263.505074721\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4800000.0)
   (*
    (-
     4.16438922228
     (/
      (+
       (/ (- (/ 130977.50649958357 x) 3655.1204654076414) x)
       110.1139242984811)
      x))
    x)
   (if (<= x 14200000000000.0)
     (/
      (* (fma (fma 137.519416416 x y) x z) (- x 2.0))
      (+
       47.066876606
       (*
        (+ 313.399215894 (* (+ (* (+ 43.3400022514 x) x) 263.505074721) x))
        x)))
     (*
      (+ (/ (- (/ 3655.1204654076414 x) 110.1139242984811) x) 4.16438922228)
      x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4800000.0) {
		tmp = (4.16438922228 - (((((130977.50649958357 / x) - 3655.1204654076414) / x) + 110.1139242984811) / x)) * x;
	} else if (x <= 14200000000000.0) {
		tmp = (fma(fma(137.519416416, x, y), x, z) * (x - 2.0)) / (47.066876606 + ((313.399215894 + ((((43.3400022514 + x) * x) + 263.505074721) * x)) * x));
	} else {
		tmp = ((((3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4800000.0)
		tmp = Float64(Float64(4.16438922228 - Float64(Float64(Float64(Float64(Float64(130977.50649958357 / x) - 3655.1204654076414) / x) + 110.1139242984811) / x)) * x);
	elseif (x <= 14200000000000.0)
		tmp = Float64(Float64(fma(fma(137.519416416, x, y), x, z) * Float64(x - 2.0)) / Float64(47.066876606 + Float64(Float64(313.399215894 + Float64(Float64(Float64(Float64(43.3400022514 + x) * x) + 263.505074721) * x)) * x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4800000.0], N[(N[(4.16438922228 - N[(N[(N[(N[(N[(130977.50649958357 / x), $MachinePrecision] - 3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + 110.1139242984811), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 14200000000000.0], N[(N[(N[(N[(137.519416416 * x + y), $MachinePrecision] * x + z), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(47.066876606 + N[(N[(313.399215894 + N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x), $MachinePrecision] + 263.505074721), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] + 4.16438922228), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8e6

   1. Initial program 11.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 11.6%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 88.6%

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

   if -4.8e6 < x < 1.42e13

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-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 99.7

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

   5. Applied rewrites 99.7%

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

   if 1.42e13 < x

   1. Initial program 18.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}}\right) - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} \]
   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 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;\left(4.16438922228 - \frac{\frac{\frac{130977.50649958357}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x}\right) \cdot x\\ \mathbf{elif}\;x \leq 14200000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(137.519416416, x, y\right), x, z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(\left(43.3400022514 + x\right) \cdot x + 263.505074721\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;\left(4.16438922228 - \frac{\frac{\frac{130977.50649958357}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x}\right) \cdot x\\ \mathbf{elif}\;x \leq 14200000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z\right) \cdot \left(x - 2\right)}{\left(\mathsf{fma}\left(x, 263.505074721, \left(\left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4800000.0)
   (*
    (-
     4.16438922228
     (/
      (+
       (/ (- (/ 130977.50649958357 x) 3655.1204654076414) x)
       110.1139242984811)
      x))
    x)
   (if (<= x 14200000000000.0)
     (/
      (* (fma y x z) (- x 2.0))
      (+
       (*
        (+ (fma x 263.505074721 (* (* (+ 43.3400022514 x) x) x)) 313.399215894)
        x)
       47.066876606))
     (*
      (+ (/ (- (/ 3655.1204654076414 x) 110.1139242984811) x) 4.16438922228)
      x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4800000.0) {
		tmp = (4.16438922228 - (((((130977.50649958357 / x) - 3655.1204654076414) / x) + 110.1139242984811) / x)) * x;
	} else if (x <= 14200000000000.0) {
		tmp = (fma(y, x, z) * (x - 2.0)) / (((fma(x, 263.505074721, (((43.3400022514 + x) * x) * x)) + 313.399215894) * x) + 47.066876606);
	} else {
		tmp = ((((3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4800000.0)
		tmp = Float64(Float64(4.16438922228 - Float64(Float64(Float64(Float64(Float64(130977.50649958357 / x) - 3655.1204654076414) / x) + 110.1139242984811) / x)) * x);
	elseif (x <= 14200000000000.0)
		tmp = Float64(Float64(fma(y, x, z) * Float64(x - 2.0)) / Float64(Float64(Float64(fma(x, 263.505074721, Float64(Float64(Float64(43.3400022514 + x) * x) * x)) + 313.399215894) * x) + 47.066876606));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4800000.0], N[(N[(4.16438922228 - N[(N[(N[(N[(N[(130977.50649958357 / x), $MachinePrecision] - 3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + 110.1139242984811), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 14200000000000.0], N[(N[(N[(y * x + z), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(x * 263.505074721 + N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + 313.399215894), $MachinePrecision] * x), $MachinePrecision] + 47.066876606), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] + 4.16438922228), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8e6

   1. Initial program 11.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 11.6%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 88.6%

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

   if -4.8e6 < x < 1.42e13

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

   if 1.42e13 < x

   1. Initial program 18.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}}\right) - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} \]
   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 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;\left(4.16438922228 - \frac{\frac{\frac{130977.50649958357}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x}\right) \cdot x\\ \mathbf{elif}\;x \leq 14200000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z\right) \cdot \left(x - 2\right)}{\left(\mathsf{fma}\left(x, 263.505074721, \left(\left(43.3400022514 + x\right) \cdot x\right) \cdot x\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;\left(4.16438922228 - \frac{\frac{\frac{130977.50649958357}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x}\right) \cdot x\\ \mathbf{elif}\;x \leq 14200000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(\left(43.3400022514 + x\right) \cdot x + 263.505074721\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4800000.0)
   (*
    (-
     4.16438922228
     (/
      (+
       (/ (- (/ 130977.50649958357 x) 3655.1204654076414) x)
       110.1139242984811)
      x))
    x)
   (if (<= x 14200000000000.0)
     (/
      (* (fma y x z) (- x 2.0))
      (+
       47.066876606
       (*
        (+ 313.399215894 (* (+ (* (+ 43.3400022514 x) x) 263.505074721) x))
        x)))
     (*
      (+ (/ (- (/ 3655.1204654076414 x) 110.1139242984811) x) 4.16438922228)
      x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4800000.0) {
		tmp = (4.16438922228 - (((((130977.50649958357 / x) - 3655.1204654076414) / x) + 110.1139242984811) / x)) * x;
	} else if (x <= 14200000000000.0) {
		tmp = (fma(y, x, z) * (x - 2.0)) / (47.066876606 + ((313.399215894 + ((((43.3400022514 + x) * x) + 263.505074721) * x)) * x));
	} else {
		tmp = ((((3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4800000.0)
		tmp = Float64(Float64(4.16438922228 - Float64(Float64(Float64(Float64(Float64(130977.50649958357 / x) - 3655.1204654076414) / x) + 110.1139242984811) / x)) * x);
	elseif (x <= 14200000000000.0)
		tmp = Float64(Float64(fma(y, x, z) * Float64(x - 2.0)) / Float64(47.066876606 + Float64(Float64(313.399215894 + Float64(Float64(Float64(Float64(43.3400022514 + x) * x) + 263.505074721) * x)) * x)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4800000.0], N[(N[(4.16438922228 - N[(N[(N[(N[(N[(130977.50649958357 / x), $MachinePrecision] - 3655.1204654076414), $MachinePrecision] / x), $MachinePrecision] + 110.1139242984811), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 14200000000000.0], N[(N[(N[(y * x + z), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(47.066876606 + N[(N[(313.399215894 + N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x), $MachinePrecision] + 263.505074721), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] + 4.16438922228), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.8e6

   1. Initial program 11.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 11.6%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 88.6%

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

   if -4.8e6 < x < 1.42e13

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.4

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

   5. Applied rewrites 97.4%

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

   if 1.42e13 < x

   1. Initial program 18.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}}\right) - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} \]
   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 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4800000:\\ \;\;\;\;\left(4.16438922228 - \frac{\frac{\frac{130977.50649958357}{x} - 3655.1204654076414}{x} + 110.1139242984811}{x}\right) \cdot x\\ \mathbf{elif}\;x \leq 14200000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z\right) \cdot \left(x - 2\right)}{47.066876606 + \left(313.399215894 + \left(\left(43.3400022514 + x\right) \cdot x + 263.505074721\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \mathbf{elif}\;x \leq 14200000000000:\\ \;\;\;\;\left(\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \left(2 - x\right)\right) \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.22e+30)
   (/ 1.0 (/ 0.24013125253755718 x))
   (if (<= x 14200000000000.0)
     (*
      (*
       (/
        -1.0
        (fma
         (fma (fma (+ 43.3400022514 x) x 263.505074721) x 313.399215894)
         x
         47.066876606))
       (- 2.0 x))
      (fma y x z))
     (*
      (+ (/ (- (/ 3655.1204654076414 x) 110.1139242984811) x) 4.16438922228)
      x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.22e+30) {
		tmp = 1.0 / (0.24013125253755718 / x);
	} else if (x <= 14200000000000.0) {
		tmp = ((-1.0 / fma(fma(fma((43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606)) * (2.0 - x)) * fma(y, x, z);
	} else {
		tmp = ((((3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.22e+30)
		tmp = Float64(1.0 / Float64(0.24013125253755718 / x));
	elseif (x <= 14200000000000.0)
		tmp = Float64(Float64(Float64(-1.0 / fma(fma(fma(Float64(43.3400022514 + x), x, 263.505074721), x, 313.399215894), x, 47.066876606)) * Float64(2.0 - x)) * fma(y, x, z));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.22e+30], N[(1.0 / N[(0.24013125253755718 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 14200000000000.0], N[(N[(N[(-1.0 / N[(N[(N[(N[(43.3400022514 + x), $MachinePrecision] * x + 263.505074721), $MachinePrecision] * x + 313.399215894), $MachinePrecision] * x + 47.066876606), $MachinePrecision]), $MachinePrecision] * N[(2.0 - x), $MachinePrecision]), $MachinePrecision] * N[(y * x + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] + 4.16438922228), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.22e30

   1. Initial program 5.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 5.6

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 5.6

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

   4. Applied rewrites 5.6%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 1.1

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

   7. Applied rewrites 1.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

   9. Applied rewrites 1.1%

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

   10. Taylor expanded in x around inf

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

       1. lower-/.f64 92.9

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

   12. Applied rewrites 92.9%

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

   if -1.22e30 < x < 1.42e13

   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

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 98.9

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 98.9

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 94.8

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

   7. Applied rewrites 94.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 95.2%

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

   if 1.42e13 < x

   1. Initial program 18.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}}\right) - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} \]
   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 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+30}:\\ \;\;\;\;\frac{1}{\frac{0.24013125253755718}{x}}\\ \mathbf{elif}\;x \leq 14200000000000:\\ \;\;\;\;\left(\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514 + x, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)} \cdot \left(2 - x\right)\right) \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \mathbf{if}\;x \leq -35:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 480000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z\right) \cdot \left(x - 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          (+
           (/ (- (/ 3655.1204654076414 x) 110.1139242984811) x)
           4.16438922228)
          x)))
   (if (<= x -35.0)
     t_0
     (if (<= x 480000000000.0)
       (/
        (* (fma y x z) (- x 2.0))
        (fma
         (fma (fma 43.3400022514 x 263.505074721) x 313.399215894)
         x
         47.066876606))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((((3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x;
	double tmp;
	if (x <= -35.0) {
		tmp = t_0;
	} else if (x <= 480000000000.0) {
		tmp = (fma(y, x, z) * (x - 2.0)) / fma(fma(fma(43.3400022514, x, 263.505074721), x, 313.399215894), x, 47.066876606);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x)
	tmp = 0.0
	if (x <= -35.0)
		tmp = t_0;
	elseif (x <= 480000000000.0)
		tmp = Float64(Float64(fma(y, x, z) * Float64(x - 2.0)) / fma(fma(fma(43.3400022514, x, 263.505074721), x, 313.399215894), x, 47.066876606));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] + 4.16438922228), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -35.0], t$95$0, If[LessEqual[x, 480000000000.0], N[(N[(N[(y * x + z), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(43.3400022514 * x + 263.505074721), $MachinePrecision] * x + 313.399215894), $MachinePrecision] * x + 47.066876606), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -35 or 4.8e11 < x

   1. Initial program 16.1%

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   if -35 < x < 4.8e11

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 95.9

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

   10. Applied rewrites 95.9%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -35:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \mathbf{elif}\;x \leq 480000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z\right) \cdot \left(x - 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(43.3400022514, x, 263.505074721\right), x, 313.399215894\right), x, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \mathbf{if}\;x \leq -40000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 480000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z\right) \cdot \left(x - 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(263.505074721, x, 313.399215894\right), x, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          (+
           (/ (- (/ 3655.1204654076414 x) 110.1139242984811) x)
           4.16438922228)
          x)))
   (if (<= x -40000.0)
     t_0
     (if (<= x 480000000000.0)
       (/
        (* (fma y x z) (- x 2.0))
        (fma (fma 263.505074721 x 313.399215894) x 47.066876606))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((((3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x;
	double tmp;
	if (x <= -40000.0) {
		tmp = t_0;
	} else if (x <= 480000000000.0) {
		tmp = (fma(y, x, z) * (x - 2.0)) / fma(fma(263.505074721, x, 313.399215894), x, 47.066876606);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x)
	tmp = 0.0
	if (x <= -40000.0)
		tmp = t_0;
	elseif (x <= 480000000000.0)
		tmp = Float64(Float64(fma(y, x, z) * Float64(x - 2.0)) / fma(fma(263.505074721, x, 313.399215894), x, 47.066876606));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] + 4.16438922228), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -40000.0], t$95$0, If[LessEqual[x, 480000000000.0], N[(N[(N[(y * x + z), $MachinePrecision] * N[(x - 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(263.505074721 * x + 313.399215894), $MachinePrecision] * x + 47.066876606), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e4 or 4.8e11 < x

   1. Initial program 15.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}}\right) - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} \]
   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 89.4

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

   5. Applied rewrites 89.4%

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

   if -4e4 < x < 4.8e11

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 95.0

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

   10. Applied rewrites 95.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40000:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \mathbf{elif}\;x \leq 480000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, x, z\right) \cdot \left(x - 2\right)}{\mathsf{fma}\left(\mathsf{fma}\left(263.505074721, x, 313.399215894\right), x, 47.066876606\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \mathbf{if}\;x \leq -40000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 480000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(10.238818846568002, x, -1.787568985856513\right), x, 0.3041881842569256\right), x, -0.0424927283095952\right) \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          (+
           (/ (- (/ 3655.1204654076414 x) 110.1139242984811) x)
           4.16438922228)
          x)))
   (if (<= x -40000.0)
     t_0
     (if (<= x 480000000000.0)
       (*
        (fma
         (fma
          (fma 10.238818846568002 x -1.787568985856513)
          x
          0.3041881842569256)
         x
         -0.0424927283095952)
        (fma y x z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((((3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x;
	double tmp;
	if (x <= -40000.0) {
		tmp = t_0;
	} else if (x <= 480000000000.0) {
		tmp = fma(fma(fma(10.238818846568002, x, -1.787568985856513), x, 0.3041881842569256), x, -0.0424927283095952) * fma(y, x, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(Float64(3655.1204654076414 / x) - 110.1139242984811) / x) + 4.16438922228) * x)
	tmp = 0.0
	if (x <= -40000.0)
		tmp = t_0;
	elseif (x <= 480000000000.0)
		tmp = Float64(fma(fma(fma(10.238818846568002, x, -1.787568985856513), x, 0.3041881842569256), x, -0.0424927283095952) * fma(y, x, z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(N[(3655.1204654076414 / x), $MachinePrecision] - 110.1139242984811), $MachinePrecision] / x), $MachinePrecision] + 4.16438922228), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -40000.0], t$95$0, If[LessEqual[x, 480000000000.0], N[(N[(N[(N[(10.238818846568002 * x + -1.787568985856513), $MachinePrecision] * x + 0.3041881842569256), $MachinePrecision] * x + -0.0424927283095952), $MachinePrecision] * N[(y * x + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e4 or 4.8e11 < x

   1. Initial program 15.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(\frac{104109730557}{25000000000} + \frac{\frac{2284450290879775841688574159837293}{625000000000000000000000000000}}{{x}^{2}}\right) - \frac{13764240537310136880149}{125000000000000000000} \cdot \frac{1}{x}\right)} \]
   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 89.4

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

   5. Applied rewrites 89.4%

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

   if -4e4 < x < 4.8e11

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 97.1%

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

   10. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 94.7

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

   12. Applied rewrites 94.7%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40000:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \mathbf{elif}\;x \leq 480000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(10.238818846568002, x, -1.787568985856513\right), x, 0.3041881842569256\right), x, -0.0424927283095952\right) \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{3655.1204654076414}{x} - 110.1139242984811}{x} + 4.16438922228\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \mathbf{if}\;x \leq -40000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 480000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(10.238818846568002, x, -1.787568985856513\right), x, 0.3041881842569256\right), x, -0.0424927283095952\right) \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 4.16438922228 (/ 110.1139242984811 x)) x)))
   (if (<= x -40000.0)
     t_0
     (if (<= x 480000000000.0)
       (*
        (fma
         (fma
          (fma 10.238818846568002 x -1.787568985856513)
          x
          0.3041881842569256)
         x
         -0.0424927283095952)
        (fma y x z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.16438922228 - (110.1139242984811 / x)) * x;
	double tmp;
	if (x <= -40000.0) {
		tmp = t_0;
	} else if (x <= 480000000000.0) {
		tmp = fma(fma(fma(10.238818846568002, x, -1.787568985856513), x, 0.3041881842569256), x, -0.0424927283095952) * fma(y, x, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.16438922228 - Float64(110.1139242984811 / x)) * x)
	tmp = 0.0
	if (x <= -40000.0)
		tmp = t_0;
	elseif (x <= 480000000000.0)
		tmp = Float64(fma(fma(fma(10.238818846568002, x, -1.787568985856513), x, 0.3041881842569256), x, -0.0424927283095952) * fma(y, x, z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.16438922228 - N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -40000.0], t$95$0, If[LessEqual[x, 480000000000.0], N[(N[(N[(N[(10.238818846568002 * x + -1.787568985856513), $MachinePrecision] * x + 0.3041881842569256), $MachinePrecision] * x + -0.0424927283095952), $MachinePrecision] * N[(y * x + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e4 or 4.8e11 < x

   1. Initial program 15.4%

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

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. neg-sub0 N/A

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if -4e4 < x < 4.8e11

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 97.1%

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

   10. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

       10. lower-fma.f64 94.7

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

   12. Applied rewrites 94.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40000:\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \mathbf{elif}\;x \leq 480000000000:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(10.238818846568002, x, -1.787568985856513\right), x, 0.3041881842569256\right), x, -0.0424927283095952\right) \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \mathbf{if}\;x \leq -40000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1.787568985856513, x, 0.3041881842569256\right), x, -0.0424927283095952\right) \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 4.16438922228 (/ 110.1139242984811 x)) x)))
   (if (<= x -40000.0)
     t_0
     (if (<= x 2.0)
       (*
        (fma
         (fma -1.787568985856513 x 0.3041881842569256)
         x
         -0.0424927283095952)
        (fma y x z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.16438922228 - (110.1139242984811 / x)) * x;
	double tmp;
	if (x <= -40000.0) {
		tmp = t_0;
	} else if (x <= 2.0) {
		tmp = fma(fma(-1.787568985856513, x, 0.3041881842569256), x, -0.0424927283095952) * fma(y, x, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.16438922228 - Float64(110.1139242984811 / x)) * x)
	tmp = 0.0
	if (x <= -40000.0)
		tmp = t_0;
	elseif (x <= 2.0)
		tmp = Float64(fma(fma(-1.787568985856513, x, 0.3041881842569256), x, -0.0424927283095952) * fma(y, x, z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.16438922228 - N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -40000.0], t$95$0, If[LessEqual[x, 2.0], N[(N[(N[(-1.787568985856513 * x + 0.3041881842569256), $MachinePrecision] * x + -0.0424927283095952), $MachinePrecision] * N[(y * x + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e4 or 2 < x

   1. Initial program 16.7%

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

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. neg-sub0 N/A

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 87.8

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

   5. Applied rewrites 87.8%

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

   if -4e4 < x < 2

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 97.1%

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

   10. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 96.0

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

   12. Applied rewrites 96.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40000:\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-1.787568985856513, x, 0.3041881842569256\right), x, -0.0424927283095952\right) \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \mathbf{if}\;x \leq -40000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 480000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right) \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- 4.16438922228 (/ 110.1139242984811 x)) x)))
   (if (<= x -40000.0)
     t_0
     (if (<= x 480000000000.0)
       (* (fma x 0.3041881842569256 -0.0424927283095952) (fma y x z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.16438922228 - (110.1139242984811 / x)) * x;
	double tmp;
	if (x <= -40000.0) {
		tmp = t_0;
	} else if (x <= 480000000000.0) {
		tmp = fma(x, 0.3041881842569256, -0.0424927283095952) * fma(y, x, z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.16438922228 - Float64(110.1139242984811 / x)) * x)
	tmp = 0.0
	if (x <= -40000.0)
		tmp = t_0;
	elseif (x <= 480000000000.0)
		tmp = Float64(fma(x, 0.3041881842569256, -0.0424927283095952) * fma(y, x, z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.16438922228 - N[(110.1139242984811 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -40000.0], t$95$0, If[LessEqual[x, 480000000000.0], N[(N[(x * 0.3041881842569256 + -0.0424927283095952), $MachinePrecision] * N[(y * x + z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e4 or 4.8e11 < x

   1. Initial program 15.4%

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

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

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

      5. associate-+l- N/A

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

      6. neg-sub0 N/A

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. lower--.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if -4e4 < x < 4.8e11

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 97.1%

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

   10. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 94.3

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

   12. Applied rewrites 94.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40000:\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \mathbf{elif}\;x \leq 480000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right) \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(4.16438922228 - \frac{110.1139242984811}{x}\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 89.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -240000:\\ \;\;\;\;4.16438922228 \cdot x\\ \mathbf{elif}\;x \leq 480000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right) \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -240000.0)
   (* 4.16438922228 x)
   (if (<= x 480000000000.0)
     (* (fma x 0.3041881842569256 -0.0424927283095952) (fma y x z))
     (* 4.16438922228 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -240000.0) {
		tmp = 4.16438922228 * x;
	} else if (x <= 480000000000.0) {
		tmp = fma(x, 0.3041881842569256, -0.0424927283095952) * fma(y, x, z);
	} else {
		tmp = 4.16438922228 * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -240000.0)
		tmp = Float64(4.16438922228 * x);
	elseif (x <= 480000000000.0)
		tmp = Float64(fma(x, 0.3041881842569256, -0.0424927283095952) * fma(y, x, z));
	else
		tmp = Float64(4.16438922228 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -240000.0], N[(4.16438922228 * x), $MachinePrecision], If[LessEqual[x, 480000000000.0], N[(N[(x * 0.3041881842569256 + -0.0424927283095952), $MachinePrecision] * N[(y * x + z), $MachinePrecision]), $MachinePrecision], N[(4.16438922228 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e5 or 4.8e11 < x

   1. Initial program 15.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if -2.4e5 < x < 4.8e11

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 97.1%

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

   10. Taylor expanded in x around 0

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

       1. sub-neg N/A

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

       2. *-commutative N/A

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

       3. metadata-eval N/A

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

       4. lower-fma.f64 94.3

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

   12. Applied rewrites 94.3%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -240000:\\ \;\;\;\;4.16438922228 \cdot x\\ \mathbf{elif}\;x \leq 480000000000:\\ \;\;\;\;\mathsf{fma}\left(x, 0.3041881842569256, -0.0424927283095952\right) \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 89.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -240000:\\ \;\;\;\;4.16438922228 \cdot x\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;-0.0424927283095952 \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -240000.0)
   (* 4.16438922228 x)
   (if (<= x 2.0) (* -0.0424927283095952 (fma y x z)) (* 4.16438922228 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -240000.0) {
		tmp = 4.16438922228 * x;
	} else if (x <= 2.0) {
		tmp = -0.0424927283095952 * fma(y, x, z);
	} else {
		tmp = 4.16438922228 * x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -240000.0)
		tmp = Float64(4.16438922228 * x);
	elseif (x <= 2.0)
		tmp = Float64(-0.0424927283095952 * fma(y, x, z));
	else
		tmp = Float64(4.16438922228 * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -240000.0], N[(4.16438922228 * x), $MachinePrecision], If[LessEqual[x, 2.0], N[(-0.0424927283095952 * N[(y * x + z), $MachinePrecision]), $MachinePrecision], N[(4.16438922228 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e5 or 2 < x

   1. Initial program 16.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 87.0

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

   5. Applied rewrites 87.0%

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

   if -2.4e5 < x < 2

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 99.7

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.4

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

   7. Applied rewrites 97.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 97.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 95.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -240000:\\ \;\;\;\;4.16438922228 \cdot x\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;-0.0424927283095952 \cdot \mathsf{fma}\left(y, x, z\right)\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 77.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -40000:\\ \;\;\;\;4.16438922228 \cdot x\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{-13}:\\ \;\;\;\;-0.0424927283095952 \cdot z\\ \mathbf{else}:\\ \;\;\;\;4.16438922228 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -40000.0)
   (* 4.16438922228 x)
   (if (<= x 1.45e-13) (* -0.0424927283095952 z) (* 4.16438922228 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -40000.0) {
		tmp = 4.16438922228 * x;
	} else if (x <= 1.45e-13) {
		tmp = -0.0424927283095952 * z;
	} else {
		tmp = 4.16438922228 * x;
	}
	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 <= (-40000.0d0)) then
        tmp = 4.16438922228d0 * x
    else if (x <= 1.45d-13) then
        tmp = (-0.0424927283095952d0) * z
    else
        tmp = 4.16438922228d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -40000.0) {
		tmp = 4.16438922228 * x;
	} else if (x <= 1.45e-13) {
		tmp = -0.0424927283095952 * z;
	} else {
		tmp = 4.16438922228 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -40000.0:
		tmp = 4.16438922228 * x
	elif x <= 1.45e-13:
		tmp = -0.0424927283095952 * z
	else:
		tmp = 4.16438922228 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -40000.0)
		tmp = Float64(4.16438922228 * x);
	elseif (x <= 1.45e-13)
		tmp = Float64(-0.0424927283095952 * z);
	else
		tmp = Float64(4.16438922228 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -40000.0)
		tmp = 4.16438922228 * x;
	elseif (x <= 1.45e-13)
		tmp = -0.0424927283095952 * z;
	else
		tmp = 4.16438922228 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -40000.0], N[(4.16438922228 * x), $MachinePrecision], If[LessEqual[x, 1.45e-13], N[(-0.0424927283095952 * z), $MachinePrecision], N[(4.16438922228 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4e4 or 1.4499999999999999e-13 < x

   1. Initial program 17.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   if -4e4 < x < 1.4499999999999999e-13

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 70.2

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

   5. Applied rewrites 70.2%

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

Alternative 17: 34.6% 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 56.6%

   \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606} \]
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 35.3

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

5. Applied rewrites 35.3%

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

Developer Target 1: 98.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024235 
(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)))