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

Percentage Accurate: 58.2% → 98.3%

Time: 17.9s

Alternatives: 18

Speedup: 11.3×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b))
   (+
    (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z)
    0.607771387771))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x + ((y * ((((((((z * 3.13060547623d0) + 11.1667541262d0) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407d0) * z) + 31.4690115749d0) * z) + 11.9400905721d0) * z) + 0.607771387771d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

Python

def code(x, y, z, t, a, b):
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))

Julia

function code(x, y, z, t, a, b)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 58.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b))
   (+
    (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z)
    0.607771387771))))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x + ((y * ((((((((z * 3.13060547623d0) + 11.1667541262d0) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407d0) * z) + 31.4690115749d0) * z) + 11.9400905721d0) * z) + 0.607771387771d0))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
}

Python

def code(x, y, z, t, a, b):
	return x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771))

Julia

function code(x, y, z, t, a, b)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x + ((y * ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)) / (((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x + N[(N[(y * N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)\\ \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), a\right), b\right)}{t\_1}, \mathsf{fma}\left(y, t \cdot \frac{z \cdot z}{t\_1}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right) + \left(\mathsf{fma}\left(y, \frac{11.1667541262}{z}, t \cdot \frac{y}{z \cdot z}\right) - \mathsf{fma}\left(y, \frac{47.69379582500642}{z}, \mathsf{fma}\left(y, \frac{98.5170599679272}{z \cdot z}, \frac{y \cdot -556.47806218377}{z \cdot z}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (fma
          z
          (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
          0.607771387771)))
   (if (<=
        (/
         (*
          y
          (+
           (* z (+ (* z (+ (* z (+ (* z 3.13060547623) 11.1667541262)) t)) a))
           b))
         (+
          (*
           z
           (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
          0.607771387771))
        INFINITY)
     (fma
      y
      (/ (fma z (fma z (* z (fma z 3.13060547623 11.1667541262)) a) b) t_1)
      (fma y (* t (/ (* z z) t_1)) x))
     (+
      (fma y 3.13060547623 x)
      (-
       (fma y (/ 11.1667541262 z) (* t (/ y (* z z))))
       (fma
        y
        (/ 47.69379582500642 z)
        (fma
         y
         (/ 98.5170599679272 (* z z))
         (/ (* y -556.47806218377) (* z z)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771);
	double tmp;
	if (((y * ((z * ((z * ((z * ((z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / ((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771)) <= ((double) INFINITY)) {
		tmp = fma(y, (fma(z, fma(z, (z * fma(z, 3.13060547623, 11.1667541262)), a), b) / t_1), fma(y, (t * ((z * z) / t_1)), x));
	} else {
		tmp = fma(y, 3.13060547623, x) + (fma(y, (11.1667541262 / z), (t * (y / (z * z)))) - fma(y, (47.69379582500642 / z), fma(y, (98.5170599679272 / (z * z)), ((y * -556.47806218377) / (z * z)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, fma(z, fma(z, Float64(z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771)) <= Inf)
		tmp = fma(y, Float64(fma(z, fma(z, Float64(z * fma(z, 3.13060547623, 11.1667541262)), a), b) / t_1), fma(y, Float64(t * Float64(Float64(z * z) / t_1)), x));
	else
		tmp = Float64(fma(y, 3.13060547623, x) + Float64(fma(y, Float64(11.1667541262 / z), Float64(t * Float64(y / Float64(z * z)))) - fma(y, Float64(47.69379582500642 / z), fma(y, Float64(98.5170599679272 / Float64(z * z)), Float64(Float64(y * -556.47806218377) / Float64(z * z))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(z * N[(z * N[(z + 15.234687407), $MachinePrecision] + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]}, If[LessEqual[N[(N[(y * N[(N[(z * N[(N[(z * N[(N[(z * N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(N[(z * N[(N[(z * N[(z + 15.234687407), $MachinePrecision]), $MachinePrecision] + 31.4690115749), $MachinePrecision]), $MachinePrecision] + 11.9400905721), $MachinePrecision]), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision], Infinity], N[(y * N[(N[(z * N[(z * N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(y * N[(t * N[(N[(z * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(y * 3.13060547623 + x), $MachinePrecision] + N[(N[(y * N[(11.1667541262 / z), $MachinePrecision] + N[(t * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(47.69379582500642 / z), $MachinePrecision] + N[(y * N[(98.5170599679272 / N[(z * z), $MachinePrecision]), $MachinePrecision] + N[(N[(y * -556.47806218377), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 90.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot \left(y \cdot {z}^{2}\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{y \cdot \left(b + z \cdot \left(a + {z}^{2} \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

   5. Simplified 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, \mathsf{fma}\left(y, \frac{z \cdot z}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} \cdot t, x\right)\right)} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{313060547623}{100000000000} \cdot y + \left(\frac{55833770631}{5000000000} \cdot \frac{y}{z} + \frac{t \cdot y}{{z}^{2}}\right)\right)\right) - \left(\frac{15234687407}{1000000000} \cdot \frac{\frac{55833770631}{5000000000} \cdot y - \frac{4769379582500641883561}{100000000000000000000} \cdot y}{{z}^{2}} + \left(\frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z} + \frac{98517059967927196814627}{1000000000000000000000} \cdot \frac{y}{{z}^{2}}\right)\right)} \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right) + \left(\mathsf{fma}\left(y, \frac{11.1667541262}{z}, t \cdot \frac{y}{z \cdot z}\right) - \mathsf{fma}\left(y, \frac{47.69379582500642}{z}, \mathsf{fma}\left(y, \frac{98.5170599679272}{z \cdot z}, \frac{y \cdot -556.47806218377}{z \cdot z}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, \mathsf{fma}\left(y, t \cdot \frac{z \cdot z}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right) + \left(\mathsf{fma}\left(y, \frac{11.1667541262}{z}, t \cdot \frac{y}{z \cdot z}\right) - \mathsf{fma}\left(y, \frac{47.69379582500642}{z}, \mathsf{fma}\left(y, \frac{98.5170599679272}{z \cdot z}, \frac{y \cdot -556.47806218377}{z \cdot z}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, t \cdot 1.6453555072203998, \mathsf{fma}\left(1.6453555072203998, a, b \cdot -32.324150453290734\right)\right), b \cdot 1.6453555072203998\right)\\ t_2 := \frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+161}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, 1.6453555072203998, b \cdot -32.324150453290734\right), \mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (*
          y
          (fma
           z
           (fma
            z
            (* t 1.6453555072203998)
            (fma 1.6453555072203998 a (* b -32.324150453290734)))
           (* b 1.6453555072203998))))
        (t_2
         (/
          (*
           y
           (+
            (* z (+ (* z (+ (* z (+ (* z 3.13060547623) 11.1667541262)) t)) a))
            b))
          (+
           (*
            z
            (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
           0.607771387771))))
   (if (<= t_2 -1e+161)
     t_1
     (if (<= t_2 1e+98)
       (fma
        z
        (* y (fma a 1.6453555072203998 (* b -32.324150453290734)))
        (fma y (* b 1.6453555072203998) x))
       (if (<= t_2 INFINITY) t_1 (fma y 3.13060547623 x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y * fma(z, fma(z, (t * 1.6453555072203998), fma(1.6453555072203998, a, (b * -32.324150453290734))), (b * 1.6453555072203998));
	double t_2 = (y * ((z * ((z * ((z * ((z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / ((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771);
	double tmp;
	if (t_2 <= -1e+161) {
		tmp = t_1;
	} else if (t_2 <= 1e+98) {
		tmp = fma(z, (y * fma(a, 1.6453555072203998, (b * -32.324150453290734))), fma(y, (b * 1.6453555072203998), x));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y * fma(z, fma(z, Float64(t * 1.6453555072203998), fma(1.6453555072203998, a, Float64(b * -32.324150453290734))), Float64(b * 1.6453555072203998)))
	t_2 = Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771))
	tmp = 0.0
	if (t_2 <= -1e+161)
		tmp = t_1;
	elseif (t_2 <= 1e+98)
		tmp = fma(z, Float64(y * fma(a, 1.6453555072203998, Float64(b * -32.324150453290734))), fma(y, Float64(b * 1.6453555072203998), x));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(z * N[(z * N[(t * 1.6453555072203998), $MachinePrecision] + N[(1.6453555072203998 * a + N[(b * -32.324150453290734), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * 1.6453555072203998), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[(z * N[(N[(z * N[(N[(z * N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(N[(z * N[(N[(z * N[(z + 15.234687407), $MachinePrecision]), $MachinePrecision] + 31.4690115749), $MachinePrecision]), $MachinePrecision] + 11.9400905721), $MachinePrecision]), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+161], t$95$1, If[LessEqual[t$95$2, 1e+98], N[(z * N[(y * N[(a * 1.6453555072203998 + N[(b * -32.324150453290734), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(b * 1.6453555072203998), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(y * 3.13060547623 + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -1e161 or 9.99999999999999998e97 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 77.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \left(\color{blue}{z \cdot \frac{a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} + \frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(z, \frac{a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}, \frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

   5. Simplified 82.0%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto y \cdot \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot b + z \cdot \left(\left(\frac{1000000000000}{607771387771} \cdot a + z \cdot \left(\frac{1000000000000}{607771387771} \cdot t - \left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} \cdot b + \left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot a + \frac{31469011574900000000000000}{369386059793087248348441} \cdot b\right)\right)\right)\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(z \cdot \left(\left(\frac{1000000000000}{607771387771} \cdot a + z \cdot \left(\frac{1000000000000}{607771387771} \cdot t - \left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} \cdot b + \left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot a + \frac{31469011574900000000000000}{369386059793087248348441} \cdot b\right)\right)\right)\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right) + \frac{1000000000000}{607771387771} \cdot b\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(z, \left(\frac{1000000000000}{607771387771} \cdot a + z \cdot \left(\frac{1000000000000}{607771387771} \cdot t - \left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} \cdot b + \left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot a + \frac{31469011574900000000000000}{369386059793087248348441} \cdot b\right)\right)\right)\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b, \frac{1000000000000}{607771387771} \cdot b\right)} \]

   8. Simplified 70.2%

      \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 1.6453555072203998 \cdot t - \mathsf{fma}\left(b, -549.8376187179895, 32.324150453290734 \cdot a\right), \mathsf{fma}\left(1.6453555072203998, a, b \cdot -32.324150453290734\right)\right), 1.6453555072203998 \cdot b\right)} \]

   9. Taylor expanded in t around inf

      \[\leadsto y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{1000000000000}{607771387771} \cdot t}, \mathsf{fma}\left(\frac{1000000000000}{607771387771}, a, b \cdot \frac{-11940090572100000000000000}{369386059793087248348441}\right)\right), \frac{1000000000000}{607771387771} \cdot b\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 70.1

          \[\leadsto y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{1.6453555072203998 \cdot t}, \mathsf{fma}\left(1.6453555072203998, a, b \cdot -32.324150453290734\right)\right), 1.6453555072203998 \cdot b\right) \]

   11. Simplified 70.1%

       \[\leadsto y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{1.6453555072203998 \cdot t}, \mathsf{fma}\left(1.6453555072203998, a, b \cdot -32.324150453290734\right)\right), 1.6453555072203998 \cdot b\right) \]

   if -1e161 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < 9.99999999999999998e97

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + z \cdot \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) + z \cdot \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right) + \left(x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot a\right) \cdot y} - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(z, \left(\frac{1000000000000}{607771387771} \cdot a\right) \cdot y - \color{blue}{\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right) \cdot y}, x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)}, x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)}, x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, y \cdot \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot a + \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)\right)}, x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, y \cdot \left(\color{blue}{a \cdot \frac{1000000000000}{607771387771}} + \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)}, x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, \mathsf{neg}\left(\color{blue}{b \cdot \frac{11940090572100000000000000}{369386059793087248348441}}\right)\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, \color{blue}{b \cdot \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441}\right)\right)}\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, \color{blue}{b \cdot \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441}\right)\right)}\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, b \cdot \color{blue}{\frac{-11940090572100000000000000}{369386059793087248348441}}\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, b \cdot \frac{-11940090572100000000000000}{369386059793087248348441}\right), \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + x}\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, b \cdot \frac{-11940090572100000000000000}{369386059793087248348441}\right), \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot b\right) \cdot y} + x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, b \cdot \frac{-11940090572100000000000000}{369386059793087248348441}\right), \color{blue}{y \cdot \left(\frac{1000000000000}{607771387771} \cdot b\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, b \cdot \frac{-11940090572100000000000000}{369386059793087248348441}\right), \color{blue}{\mathsf{fma}\left(y, \frac{1000000000000}{607771387771} \cdot b, x\right)}\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, b \cdot \frac{-11940090572100000000000000}{369386059793087248348441}\right), \mathsf{fma}\left(y, \color{blue}{b \cdot \frac{1000000000000}{607771387771}}, x\right)\right) \]

      20. lower-*.f64 83.2

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, 1.6453555072203998, b \cdot -32.324150453290734\right), \mathsf{fma}\left(y, \color{blue}{b \cdot 1.6453555072203998}, x\right)\right) \]

   5. Simplified 83.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, 1.6453555072203998, b \cdot -32.324150453290734\right), \mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\right)} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 97.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq -1 \cdot 10^{+161}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, t \cdot 1.6453555072203998, \mathsf{fma}\left(1.6453555072203998, a, b \cdot -32.324150453290734\right)\right), b \cdot 1.6453555072203998\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, 1.6453555072203998, b \cdot -32.324150453290734\right), \mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, t \cdot 1.6453555072203998, \mathsf{fma}\left(1.6453555072203998, a, b \cdot -32.324150453290734\right)\right), b \cdot 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+144}:\\ \;\;\;\;1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+57}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;b \cdot \left(y \cdot 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (/
          (*
           y
           (+
            (* z (+ (* z (+ (* z (+ (* z 3.13060547623) 11.1667541262)) t)) a))
            b))
          (+
           (*
            z
            (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
           0.607771387771))))
   (if (<= t_1 -1e+144)
     (* 1.6453555072203998 (* y b))
     (if (<= t_1 2e+57)
       x
       (if (<= t_1 INFINITY)
         (* b (* y 1.6453555072203998))
         (fma y 3.13060547623 x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y * ((z * ((z * ((z * ((z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / ((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771);
	double tmp;
	if (t_1 <= -1e+144) {
		tmp = 1.6453555072203998 * (y * b);
	} else if (t_1 <= 2e+57) {
		tmp = x;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = b * (y * 1.6453555072203998);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771))
	tmp = 0.0
	if (t_1 <= -1e+144)
		tmp = Float64(1.6453555072203998 * Float64(y * b));
	elseif (t_1 <= 2e+57)
		tmp = x;
	elseif (t_1 <= Inf)
		tmp = Float64(b * Float64(y * 1.6453555072203998));
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * N[(N[(z * N[(N[(z * N[(N[(z * N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(N[(z * N[(N[(z * N[(z + 15.234687407), $MachinePrecision]), $MachinePrecision] + 31.4690115749), $MachinePrecision]), $MachinePrecision] + 11.9400905721), $MachinePrecision]), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+144], N[(1.6453555072203998 * N[(y * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+57], x, If[LessEqual[t$95$1, Infinity], N[(b * N[(y * 1.6453555072203998), $MachinePrecision]), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -1.00000000000000002e144

   1. Initial program 77.9%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \left(\color{blue}{z \cdot \frac{a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} + \frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(z, \frac{a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}, \frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

   5. Simplified 87.2%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto y \cdot \color{blue}{\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \frac{b}{\color{blue}{z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right) + \frac{607771387771}{1000000000000}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{b}{\color{blue}{\mathsf{fma}\left(z, \frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right), \frac{607771387771}{1000000000000}\right)}} \]

      4. +-commutative N/A

         \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right) + \frac{119400905721}{10000000000}}, \frac{607771387771}{1000000000000}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right), \frac{119400905721}{10000000000}\right)}, \frac{607771387771}{1000000000000}\right)} \]

      6. +-commutative N/A

         \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{15234687407}{1000000000} + z\right) + \frac{314690115749}{10000000000}}, \frac{119400905721}{10000000000}\right), \frac{607771387771}{1000000000000}\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{15234687407}{1000000000} + z, \frac{314690115749}{10000000000}\right)}, \frac{119400905721}{10000000000}\right), \frac{607771387771}{1000000000000}\right)} \]

      8. +-commutative N/A

         \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z + \frac{15234687407}{1000000000}}, \frac{314690115749}{10000000000}\right), \frac{119400905721}{10000000000}\right), \frac{607771387771}{1000000000000}\right)} \]

      9. lower-+.f64 62.8

         \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z + 15.234687407}, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} \]

   8. Simplified 62.8%

      \[\leadsto y \cdot \color{blue}{\frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}} \]

   9. Taylor expanded in z around inf

      \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \color{blue}{{z}^{3}}, \frac{607771387771}{1000000000000}\right)} \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(z \cdot z\right)}, \frac{607771387771}{1000000000000}\right)} \]

       2. unpow2 N/A

          \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, z \cdot \color{blue}{{z}^{2}}, \frac{607771387771}{1000000000000}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \color{blue}{z \cdot {z}^{2}}, \frac{607771387771}{1000000000000}\right)} \]

       4. unpow2 N/A

          \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, z \cdot \color{blue}{\left(z \cdot z\right)}, \frac{607771387771}{1000000000000}\right)} \]

       5. lower-*.f64 62.8

          \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, z \cdot \color{blue}{\left(z \cdot z\right)}, 0.607771387771\right)} \]

   11. Simplified 62.8%

       \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(z \cdot z\right)}, 0.607771387771\right)} \]

   12. Taylor expanded in z around 0

       \[\leadsto \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1000000000000}{607771387771} \cdot \color{blue}{\left(y \cdot b\right)} \]

       3. lower-*.f64 62.5

          \[\leadsto 1.6453555072203998 \cdot \color{blue}{\left(y \cdot b\right)} \]

   14. Simplified 62.5%

       \[\leadsto \color{blue}{1.6453555072203998 \cdot \left(y \cdot b\right)} \]

   if -1.00000000000000002e144 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < 2.0000000000000001e57

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 65.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 65.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot x}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot \color{blue}{\left(-1 \cdot x\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot \left(-1 \cdot x\right)} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(-1 \cdot x\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{y}{x} \cdot \frac{-313060547623}{100000000000}} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot x\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{y}{x} \cdot \frac{-313060547623}{100000000000} + \color{blue}{-1}\right) \cdot \left(-1 \cdot x\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x}, \frac{-313060547623}{100000000000}, -1\right)} \cdot \left(-1 \cdot x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{x}}, \frac{-313060547623}{100000000000}, -1\right) \cdot \left(-1 \cdot x\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{x}, \frac{-313060547623}{100000000000}, -1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      12. lower-neg.f64 65.0

          \[\leadsto \mathsf{fma}\left(\frac{y}{x}, -3.13060547623, -1\right) \cdot \color{blue}{\left(-x\right)} \]

   8. Simplified 65.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x}, -3.13060547623, -1\right) \cdot \left(-x\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1} \cdot \left(\mathsf{neg}\left(x\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 65.9%

       \[\leadsto \color{blue}{-1} \cdot \left(-x\right) \]
   12. Step-by-step derivation

       1. distribute-rgt-neg-out N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(-1 \cdot x\right)} \]

       2. neg-mul-1 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

       3. remove-double-neg 65.9

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

   13. Applied egg-rr 65.9%

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

   if 2.0000000000000001e57 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 80.8%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + x} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot b\right) \cdot y} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{1000000000000}{607771387771} \cdot b\right)} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1000000000000}{607771387771} \cdot b, x\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{b \cdot \frac{1000000000000}{607771387771}}, x\right) \]

      6. lower-*.f64 61.3

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{b \cdot 1.6453555072203998}, x\right) \]

   5. Simplified 61.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\frac{1000000000000}{607771387771} \cdot y + \frac{x}{b}\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{b \cdot \left(\frac{1000000000000}{607771387771} \cdot y + \frac{x}{b}\right)} \]

      2. *-commutative N/A

         \[\leadsto b \cdot \left(\color{blue}{y \cdot \frac{1000000000000}{607771387771}} + \frac{x}{b}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto b \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1000000000000}{607771387771}, \frac{x}{b}\right)} \]

      4. lower-/.f64 56.6

         \[\leadsto b \cdot \mathsf{fma}\left(y, 1.6453555072203998, \color{blue}{\frac{x}{b}}\right) \]

   8. Simplified 56.6%

      \[\leadsto \color{blue}{b \cdot \mathsf{fma}\left(y, 1.6453555072203998, \frac{x}{b}\right)} \]

   9. Taylor expanded in y around inf

      \[\leadsto b \cdot \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot y\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto b \cdot \color{blue}{\left(y \cdot \frac{1000000000000}{607771387771}\right)} \]

       2. lower-*.f64 46.7

          \[\leadsto b \cdot \color{blue}{\left(y \cdot 1.6453555072203998\right)} \]

   11. Simplified 46.7%

       \[\leadsto b \cdot \color{blue}{\left(y \cdot 1.6453555072203998\right)} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 97.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq -1 \cdot 10^{+144}:\\ \;\;\;\;1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq 2 \cdot 10^{+57}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;b \cdot \left(y \cdot 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1.6453555072203998 \cdot \left(y \cdot b\right)\\ t_2 := \frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+57}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 1.6453555072203998 (* y b)))
        (t_2
         (/
          (*
           y
           (+
            (* z (+ (* z (+ (* z (+ (* z 3.13060547623) 11.1667541262)) t)) a))
            b))
          (+
           (*
            z
            (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
           0.607771387771))))
   (if (<= t_2 -1e+144)
     t_1
     (if (<= t_2 2e+57)
       x
       (if (<= t_2 INFINITY) t_1 (fma y 3.13060547623 x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = 1.6453555072203998 * (y * b);
	double t_2 = (y * ((z * ((z * ((z * ((z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / ((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771);
	double tmp;
	if (t_2 <= -1e+144) {
		tmp = t_1;
	} else if (t_2 <= 2e+57) {
		tmp = x;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(1.6453555072203998 * Float64(y * b))
	t_2 = Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771))
	tmp = 0.0
	if (t_2 <= -1e+144)
		tmp = t_1;
	elseif (t_2 <= 2e+57)
		tmp = x;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(1.6453555072203998 * N[(y * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[(z * N[(N[(z * N[(N[(z * N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(N[(z * N[(N[(z * N[(z + 15.234687407), $MachinePrecision]), $MachinePrecision] + 31.4690115749), $MachinePrecision]), $MachinePrecision] + 11.9400905721), $MachinePrecision]), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+144], t$95$1, If[LessEqual[t$95$2, 2e+57], x, If[LessEqual[t$95$2, Infinity], t$95$1, N[(y * 3.13060547623 + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -1.00000000000000002e144 or 2.0000000000000001e57 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 79.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \left(\color{blue}{z \cdot \frac{a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} + \frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(z, \frac{a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}, \frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

   5. Simplified 80.8%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto y \cdot \color{blue}{\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \frac{b}{\color{blue}{z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right) + \frac{607771387771}{1000000000000}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{b}{\color{blue}{\mathsf{fma}\left(z, \frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right), \frac{607771387771}{1000000000000}\right)}} \]

      4. +-commutative N/A

         \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right) + \frac{119400905721}{10000000000}}, \frac{607771387771}{1000000000000}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right), \frac{119400905721}{10000000000}\right)}, \frac{607771387771}{1000000000000}\right)} \]

      6. +-commutative N/A

         \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{15234687407}{1000000000} + z\right) + \frac{314690115749}{10000000000}}, \frac{119400905721}{10000000000}\right), \frac{607771387771}{1000000000000}\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{15234687407}{1000000000} + z, \frac{314690115749}{10000000000}\right)}, \frac{119400905721}{10000000000}\right), \frac{607771387771}{1000000000000}\right)} \]

      8. +-commutative N/A

         \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z + \frac{15234687407}{1000000000}}, \frac{314690115749}{10000000000}\right), \frac{119400905721}{10000000000}\right), \frac{607771387771}{1000000000000}\right)} \]

      9. lower-+.f64 54.3

         \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z + 15.234687407}, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} \]

   8. Simplified 54.3%

      \[\leadsto y \cdot \color{blue}{\frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}} \]

   9. Taylor expanded in z around inf

      \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \color{blue}{{z}^{3}}, \frac{607771387771}{1000000000000}\right)} \]
   10. Step-by-step derivation

       1. cube-mult N/A

          \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(z \cdot z\right)}, \frac{607771387771}{1000000000000}\right)} \]

       2. unpow2 N/A

          \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, z \cdot \color{blue}{{z}^{2}}, \frac{607771387771}{1000000000000}\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \color{blue}{z \cdot {z}^{2}}, \frac{607771387771}{1000000000000}\right)} \]

       4. unpow2 N/A

          \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, z \cdot \color{blue}{\left(z \cdot z\right)}, \frac{607771387771}{1000000000000}\right)} \]

       5. lower-*.f64 54.3

          \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, z \cdot \color{blue}{\left(z \cdot z\right)}, 0.607771387771\right)} \]

   11. Simplified 54.3%

       \[\leadsto y \cdot \frac{b}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(z \cdot z\right)}, 0.607771387771\right)} \]

   12. Taylor expanded in z around 0

       \[\leadsto \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]

       2. *-commutative N/A

          \[\leadsto \frac{1000000000000}{607771387771} \cdot \color{blue}{\left(y \cdot b\right)} \]

       3. lower-*.f64 53.6

          \[\leadsto 1.6453555072203998 \cdot \color{blue}{\left(y \cdot b\right)} \]

   14. Simplified 53.6%

       \[\leadsto \color{blue}{1.6453555072203998 \cdot \left(y \cdot b\right)} \]

   if -1.00000000000000002e144 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < 2.0000000000000001e57

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 65.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 65.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot x}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot \color{blue}{\left(-1 \cdot x\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot \left(-1 \cdot x\right)} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(-1 \cdot x\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{y}{x} \cdot \frac{-313060547623}{100000000000}} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot x\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{y}{x} \cdot \frac{-313060547623}{100000000000} + \color{blue}{-1}\right) \cdot \left(-1 \cdot x\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x}, \frac{-313060547623}{100000000000}, -1\right)} \cdot \left(-1 \cdot x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{x}}, \frac{-313060547623}{100000000000}, -1\right) \cdot \left(-1 \cdot x\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{x}, \frac{-313060547623}{100000000000}, -1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      12. lower-neg.f64 65.0

          \[\leadsto \mathsf{fma}\left(\frac{y}{x}, -3.13060547623, -1\right) \cdot \color{blue}{\left(-x\right)} \]

   8. Simplified 65.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x}, -3.13060547623, -1\right) \cdot \left(-x\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1} \cdot \left(\mathsf{neg}\left(x\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 65.9%

       \[\leadsto \color{blue}{-1} \cdot \left(-x\right) \]
   12. Step-by-step derivation

       1. distribute-rgt-neg-out N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(-1 \cdot x\right)} \]

       2. neg-mul-1 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

       3. remove-double-neg 65.9

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

   13. Applied egg-rr 65.9%

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

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 97.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq -1 \cdot 10^{+144}:\\ \;\;\;\;1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq 2 \cdot 10^{+57}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)\\ \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), a\right), b\right)}{t\_1}, \mathsf{fma}\left(y, t \cdot \frac{z \cdot z}{t\_1}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (fma
          z
          (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
          0.607771387771)))
   (if (<=
        (/
         (*
          y
          (+
           (* z (+ (* z (+ (* z (+ (* z 3.13060547623) 11.1667541262)) t)) a))
           b))
         (+
          (*
           z
           (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
          0.607771387771))
        INFINITY)
     (fma
      y
      (/ (fma z (fma z (* z (fma z 3.13060547623 11.1667541262)) a) b) t_1)
      (fma y (* t (/ (* z z) t_1)) x))
     (fma y 3.13060547623 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771);
	double tmp;
	if (((y * ((z * ((z * ((z * ((z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / ((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771)) <= ((double) INFINITY)) {
		tmp = fma(y, (fma(z, fma(z, (z * fma(z, 3.13060547623, 11.1667541262)), a), b) / t_1), fma(y, (t * ((z * z) / t_1)), x));
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, fma(z, fma(z, Float64(z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771)) <= Inf)
		tmp = fma(y, Float64(fma(z, fma(z, Float64(z * fma(z, 3.13060547623, 11.1667541262)), a), b) / t_1), fma(y, Float64(t * Float64(Float64(z * z) / t_1)), x));
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(z * N[(z * N[(z + 15.234687407), $MachinePrecision] + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]}, If[LessEqual[N[(N[(y * N[(N[(z * N[(N[(z * N[(N[(z * N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(N[(z * N[(N[(z * N[(z + 15.234687407), $MachinePrecision]), $MachinePrecision] + 31.4690115749), $MachinePrecision]), $MachinePrecision] + 11.9400905721), $MachinePrecision]), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision], Infinity], N[(y * N[(N[(z * N[(z * N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(y * N[(t * N[(N[(z * z), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 90.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \left(\frac{t \cdot \left(y \cdot {z}^{2}\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{y \cdot \left(b + z \cdot \left(a + {z}^{2} \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

   5. Simplified 97.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, \mathsf{fma}\left(y, \frac{z \cdot z}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} \cdot t, x\right)\right)} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 97.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, \mathsf{fma}\left(y, t \cdot \frac{z \cdot z}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<=
      (/
       (*
        y
        (+
         (* z (+ (* z (+ (* z (+ (* z 3.13060547623) 11.1667541262)) t)) a))
         b))
       (+
        (*
         z
         (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
        0.607771387771))
      INFINITY)
   (/
    1.0
    (/
     1.0
     (fma
      (fma z (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a) b)
      (/
       y
       (fma
        z
        (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
        0.607771387771))
      x)))
   (fma y 3.13060547623 x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * ((z * ((z * ((z * ((z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / ((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771)) <= ((double) INFINITY)) {
		tmp = 1.0 / (1.0 / fma(fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b), (y / fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)), x));
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771)) <= Inf)
		tmp = Float64(1.0 / Float64(1.0 / fma(fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b), Float64(y / fma(z, fma(z, fma(z, Float64(z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)), x)));
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(y * N[(N[(z * N[(N[(z * N[(N[(z * N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(N[(z * N[(N[(z * N[(z + 15.234687407), $MachinePrecision]), $MachinePrecision] + 31.4690115749), $MachinePrecision]), $MachinePrecision] + 11.9400905721), $MachinePrecision]), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision], Infinity], N[(1.0 / N[(1.0 / N[(N[(z * N[(z * N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision] * N[(y / N[(z * N[(z * N[(z * N[(z + 15.234687407), $MachinePrecision] + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 90.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   4. Applied egg-rr 91.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)}}} \]

   if +inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64)))

   1. Initial program 0.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 97.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 97.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.8e+46)
   (fma y 3.13060547623 x)
   (if (<= z 1.08e+33)
     (fma
      (/
       1.0
       (fma
        z
        (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
        0.607771387771))
      (* y (fma z (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a) b))
      x)
     (fma y 3.13060547623 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.8e+46) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.08e+33) {
		tmp = fma((1.0 / fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)), (y * fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b)), x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.8e+46)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.08e+33)
		tmp = fma(Float64(1.0 / fma(z, fma(z, fma(z, Float64(z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)), Float64(y * fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b)), x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.8e+46], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.08e+33], N[(N[(1.0 / N[(z * N[(z * N[(z * N[(z + 15.234687407), $MachinePrecision] + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision] * N[(y * N[(z * N[(z * N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7999999999999999e46 or 1.08000000000000005e33 < z

   1. Initial program 8.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 92.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   if -3.7999999999999999e46 < z < 1.08000000000000005e33

   1. Initial program 96.9%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   4. Applied egg-rr 96.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+32}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, t, a\right), b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -6.2e+25)
   (fma y 3.13060547623 x)
   (if (<= z 6.8e+32)
     (+
      x
      (/
       (* y (fma z (fma z t a) b))
       (+
        (*
         z
         (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
        0.607771387771)))
     (fma y 3.13060547623 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -6.2e+25) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 6.8e+32) {
		tmp = x + ((y * fma(z, fma(z, t, a), b)) / ((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771));
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -6.2e+25)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 6.8e+32)
		tmp = Float64(x + Float64(Float64(y * fma(z, fma(z, t, a), b)) / Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771)));
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -6.2e+25], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 6.8e+32], N[(x + N[(N[(y * N[(z * N[(z * t + a), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(N[(z * N[(N[(z * N[(z + 15.234687407), $MachinePrecision]), $MachinePrecision] + 31.4690115749), $MachinePrecision]), $MachinePrecision] + 11.9400905721), $MachinePrecision]), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.1999999999999996e25 or 6.79999999999999957e32 < z

   1. Initial program 11.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 91.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 91.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   if -6.1999999999999996e25 < z < 6.79999999999999957e32

   1. Initial program 96.8%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto x + \frac{y \cdot \color{blue}{\left(b + z \cdot \left(a + t \cdot z\right)\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(z \cdot \left(a + t \cdot z\right) + b\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      2. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(z, a + t \cdot z, b\right)}}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      3. +-commutative N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(z, \color{blue}{t \cdot z + a}, b\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      4. *-commutative N/A

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(z, \color{blue}{z \cdot t} + a, b\right)}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \]

      5. lower-fma.f64 96.1

         \[\leadsto x + \frac{y \cdot \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, t, a\right)}, b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]

   5. Simplified 96.1%

      \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, t, a\right), b\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+32}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, t, a\right), b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), z \cdot z, b\right)}{0.607771387771}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -6.2e+16)
   (fma y 3.13060547623 x)
   (if (<= z 1.3e+17)
     (fma
      y
      (/
       (fma (fma z (fma z 3.13060547623 11.1667541262) t) (* z z) b)
       0.607771387771)
      x)
     (fma y 3.13060547623 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -6.2e+16) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.3e+17) {
		tmp = fma(y, (fma(fma(z, fma(z, 3.13060547623, 11.1667541262), t), (z * z), b) / 0.607771387771), x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -6.2e+16)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.3e+17)
		tmp = fma(y, Float64(fma(fma(z, fma(z, 3.13060547623, 11.1667541262), t), Float64(z * z), b) / 0.607771387771), x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -6.2e+16], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.3e+17], N[(y * N[(N[(N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision] + t), $MachinePrecision] * N[(z * z), $MachinePrecision] + b), $MachinePrecision] / 0.607771387771), $MachinePrecision] + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.2e16 or 1.3e17 < z

   1. Initial program 13.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 89.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 89.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   if -6.2e16 < z < 1.3e17

   1. Initial program 96.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(b + {z}^{2} \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(b + {z}^{2} \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{b + {z}^{2} \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{b + {z}^{2} \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}, x\right)} \]

   5. Simplified 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), z \cdot z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{313060547623}{100000000000}, \frac{55833770631}{5000000000}\right), t\right), z \cdot z, b\right)}{\color{blue}{\frac{607771387771}{1000000000000}}}, x\right) \]
   7. Step-by-step derivation

   8. Simplified 83.1%

      \[\leadsto \mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), z \cdot z, b\right)}{\color{blue}{0.607771387771}}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 86.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(1.6453555072203998, t, b \cdot 549.8376187179895\right), b \cdot -32.324150453290734\right), b \cdot 1.6453555072203998\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -6.3e+16)
   (fma y 3.13060547623 x)
   (if (<= z 7.5e+17)
     (fma
      y
      (fma
       z
       (fma
        z
        (fma 1.6453555072203998 t (* b 549.8376187179895))
        (* b -32.324150453290734))
       (* b 1.6453555072203998))
      x)
     (fma y 3.13060547623 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -6.3e+16) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 7.5e+17) {
		tmp = fma(y, fma(z, fma(z, fma(1.6453555072203998, t, (b * 549.8376187179895)), (b * -32.324150453290734)), (b * 1.6453555072203998)), x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -6.3e+16)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 7.5e+17)
		tmp = fma(y, fma(z, fma(z, fma(1.6453555072203998, t, Float64(b * 549.8376187179895)), Float64(b * -32.324150453290734)), Float64(b * 1.6453555072203998)), x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -6.3e+16], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 7.5e+17], N[(y * N[(z * N[(z * N[(1.6453555072203998 * t + N[(b * 549.8376187179895), $MachinePrecision]), $MachinePrecision] + N[(b * -32.324150453290734), $MachinePrecision]), $MachinePrecision] + N[(b * 1.6453555072203998), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.3e16 or 7.5e17 < z

   1. Initial program 13.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 89.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 89.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   if -6.3e16 < z < 7.5e17

   1. Initial program 96.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(b + {z}^{2} \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(b + {z}^{2} \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{b + {z}^{2} \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{b + {z}^{2} \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}, x\right)} \]

   5. Simplified 84.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), z \cdot z, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{1000000000000}{607771387771} \cdot b + z \cdot \left(z \cdot \left(\frac{1000000000000}{607771387771} \cdot t - \left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} \cdot b + \frac{31469011574900000000000000}{369386059793087248348441} \cdot b\right)\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)}, x\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{z \cdot \left(z \cdot \left(\frac{1000000000000}{607771387771} \cdot t - \left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} \cdot b + \frac{31469011574900000000000000}{369386059793087248348441} \cdot b\right)\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right) + \frac{1000000000000}{607771387771} \cdot b}, x\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(z, z \cdot \left(\frac{1000000000000}{607771387771} \cdot t - \left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} \cdot b + \frac{31469011574900000000000000}{369386059793087248348441} \cdot b\right)\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b, \frac{1000000000000}{607771387771} \cdot b\right)}, x\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1000000000000}{607771387771} \cdot t - \left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} \cdot b + \frac{31469011574900000000000000}{369386059793087248348441} \cdot b\right)\right) + \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)}, \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1000000000000}{607771387771} \cdot t - \left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} \cdot b + \frac{31469011574900000000000000}{369386059793087248348441} \cdot b\right), \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)}, \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{1000000000000}{607771387771} \cdot t + \left(\mathsf{neg}\left(\left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} \cdot b + \frac{31469011574900000000000000}{369386059793087248348441} \cdot b\right)\right)\right)}, \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right), \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(\frac{1000000000000}{607771387771}, t, \mathsf{neg}\left(\left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} \cdot b + \frac{31469011574900000000000000}{369386059793087248348441} \cdot b\right)\right)\right)}, \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right), \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

      7. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{1000000000000}{607771387771}, t, \mathsf{neg}\left(\color{blue}{b \cdot \left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} + \frac{31469011574900000000000000}{369386059793087248348441}\right)}\right)\right), \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right), \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

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

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{1000000000000}{607771387771}, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} + \frac{31469011574900000000000000}{369386059793087248348441}\right)\right)\right)}\right), \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right), \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{1000000000000}{607771387771}, t, \color{blue}{b \cdot \left(\mathsf{neg}\left(\left(\frac{-142565762869951305298410000000000000000}{224502278183706222041215714334315011} + \frac{31469011574900000000000000}{369386059793087248348441}\right)\right)\right)}\right), \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right), \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{1000000000000}{607771387771}, t, b \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{-123439798033292669987862100000000000000}{224502278183706222041215714334315011}}\right)\right)\right), \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right), \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{1000000000000}{607771387771}, t, b \cdot \color{blue}{\frac{123439798033292669987862100000000000000}{224502278183706222041215714334315011}}\right), \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right), \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{1000000000000}{607771387771}, t, b \cdot \frac{123439798033292669987862100000000000000}{224502278183706222041215714334315011}\right), \mathsf{neg}\left(\color{blue}{b \cdot \frac{11940090572100000000000000}{369386059793087248348441}}\right)\right), \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

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

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{1000000000000}{607771387771}, t, b \cdot \frac{123439798033292669987862100000000000000}{224502278183706222041215714334315011}\right), \color{blue}{b \cdot \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441}\right)\right)}\right), \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{1000000000000}{607771387771}, t, b \cdot \frac{123439798033292669987862100000000000000}{224502278183706222041215714334315011}\right), b \cdot \color{blue}{\frac{-11940090572100000000000000}{369386059793087248348441}}\right), \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(\frac{1000000000000}{607771387771}, t, b \cdot \frac{123439798033292669987862100000000000000}{224502278183706222041215714334315011}\right), \color{blue}{b \cdot \frac{-11940090572100000000000000}{369386059793087248348441}}\right), \frac{1000000000000}{607771387771} \cdot b\right), x\right) \]

      16. lower-*.f64 82.4

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(1.6453555072203998, t, b \cdot 549.8376187179895\right), b \cdot -32.324150453290734\right), \color{blue}{1.6453555072203998 \cdot b}\right), x\right) \]

   8. Simplified 82.4%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(1.6453555072203998, t, b \cdot 549.8376187179895\right), b \cdot -32.324150453290734\right), 1.6453555072203998 \cdot b\right)}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(1.6453555072203998, t, b \cdot 549.8376187179895\right), b \cdot -32.324150453290734\right), b \cdot 1.6453555072203998\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-296}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, 1.6453555072203998, b \cdot -32.324150453290734\right), \mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1e+19)
   (fma y 3.13060547623 x)
   (if (<= z 3.4e-296)
     (fma
      z
      (* y (fma a 1.6453555072203998 (* b -32.324150453290734)))
      (fma y (* b 1.6453555072203998) x))
     (if (<= z 1.06e+17)
       (fma (* y b) 1.6453555072203998 x)
       (fma y 3.13060547623 x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1e+19) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 3.4e-296) {
		tmp = fma(z, (y * fma(a, 1.6453555072203998, (b * -32.324150453290734))), fma(y, (b * 1.6453555072203998), x));
	} else if (z <= 1.06e+17) {
		tmp = fma((y * b), 1.6453555072203998, x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1e+19)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 3.4e-296)
		tmp = fma(z, Float64(y * fma(a, 1.6453555072203998, Float64(b * -32.324150453290734))), fma(y, Float64(b * 1.6453555072203998), x));
	elseif (z <= 1.06e+17)
		tmp = fma(Float64(y * b), 1.6453555072203998, x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1e+19], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 3.4e-296], N[(z * N[(y * N[(a * 1.6453555072203998 + N[(b * -32.324150453290734), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(b * 1.6453555072203998), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.06e+17], N[(N[(y * b), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1e19 or 1.06e17 < z

   1. Initial program 12.6%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 90.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 90.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   if -1e19 < z < 3.39999999999999997e-296

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + z \cdot \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) + z \cdot \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right) + \left(x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right)} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right)} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot a\right) \cdot y} - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(z, \left(\frac{1000000000000}{607771387771} \cdot a\right) \cdot y - \color{blue}{\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right) \cdot y}, x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)}, x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{y \cdot \left(\frac{1000000000000}{607771387771} \cdot a - \frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)}, x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(z, y \cdot \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot a + \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)\right)}, x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, y \cdot \left(\color{blue}{a \cdot \frac{1000000000000}{607771387771}} + \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \color{blue}{\mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, \mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441} \cdot b\right)\right)}, x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, \mathsf{neg}\left(\color{blue}{b \cdot \frac{11940090572100000000000000}{369386059793087248348441}}\right)\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, \color{blue}{b \cdot \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441}\right)\right)}\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, \color{blue}{b \cdot \left(\mathsf{neg}\left(\frac{11940090572100000000000000}{369386059793087248348441}\right)\right)}\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, b \cdot \color{blue}{\frac{-11940090572100000000000000}{369386059793087248348441}}\right), x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)\right) \]

      15. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, b \cdot \frac{-11940090572100000000000000}{369386059793087248348441}\right), \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + x}\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, b \cdot \frac{-11940090572100000000000000}{369386059793087248348441}\right), \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot b\right) \cdot y} + x\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, b \cdot \frac{-11940090572100000000000000}{369386059793087248348441}\right), \color{blue}{y \cdot \left(\frac{1000000000000}{607771387771} \cdot b\right)} + x\right) \]

      18. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, b \cdot \frac{-11940090572100000000000000}{369386059793087248348441}\right), \color{blue}{\mathsf{fma}\left(y, \frac{1000000000000}{607771387771} \cdot b, x\right)}\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, \frac{1000000000000}{607771387771}, b \cdot \frac{-11940090572100000000000000}{369386059793087248348441}\right), \mathsf{fma}\left(y, \color{blue}{b \cdot \frac{1000000000000}{607771387771}}, x\right)\right) \]

      20. lower-*.f64 82.8

          \[\leadsto \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, 1.6453555072203998, b \cdot -32.324150453290734\right), \mathsf{fma}\left(y, \color{blue}{b \cdot 1.6453555072203998}, x\right)\right) \]

   5. Simplified 82.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(a, 1.6453555072203998, b \cdot -32.324150453290734\right), \mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\right)} \]

   if 3.39999999999999997e-296 < z < 1.06e17

   1. Initial program 93.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + x} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot b\right) \cdot y} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{1000000000000}{607771387771} \cdot b\right)} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1000000000000}{607771387771} \cdot b, x\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{b \cdot \frac{1000000000000}{607771387771}}, x\right) \]

      6. lower-*.f64 77.0

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{b \cdot 1.6453555072203998}, x\right) \]

   5. Simplified 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(y \cdot b\right) \cdot \frac{1000000000000}{607771387771}} + x \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot b, \frac{1000000000000}{607771387771}, x\right)} \]

      3. lower-*.f64 77.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot b}, 1.6453555072203998, x\right) \]

   7. Applied egg-rr 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 83.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -36.52704169880642, \mathsf{fma}\left(y, 3.13060547623, x\right)\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.3e-19)
   (fma (/ y z) -36.52704169880642 (fma y 3.13060547623 x))
   (if (<= z 1.06e+17)
     (fma (* y b) 1.6453555072203998 x)
     (fma y 3.13060547623 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.3e-19) {
		tmp = fma((y / z), -36.52704169880642, fma(y, 3.13060547623, x));
	} else if (z <= 1.06e+17) {
		tmp = fma((y * b), 1.6453555072203998, x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.3e-19)
		tmp = fma(Float64(y / z), -36.52704169880642, fma(y, 3.13060547623, x));
	elseif (z <= 1.06e+17)
		tmp = fma(Float64(y * b), 1.6453555072203998, x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.3e-19], N[(N[(y / z), $MachinePrecision] * -36.52704169880642 + N[(y * 3.13060547623 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.06e+17], N[(N[(y * b), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.30000000000000006e-19

   1. Initial program 19.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{313060547623}{100000000000} \cdot y + \frac{55833770631}{5000000000} \cdot \frac{y}{z}\right)\right) - \frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{x + \left(\left(\frac{313060547623}{100000000000} \cdot y + \frac{55833770631}{5000000000} \cdot \frac{y}{z}\right) - \frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{313060547623}{100000000000} \cdot y + \frac{55833770631}{5000000000} \cdot \frac{y}{z}\right) - \frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z}\right) + x} \]

      3. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{313060547623}{100000000000} \cdot y + \left(\frac{55833770631}{5000000000} \cdot \frac{y}{z} - \frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z}\right)\right)} + x \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{55833770631}{5000000000} \cdot \frac{y}{z} - \frac{4769379582500641883561}{100000000000000000000} \cdot \frac{y}{z}\right) + \frac{313060547623}{100000000000} \cdot y\right)} + x \]

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

         \[\leadsto \left(\color{blue}{\frac{y}{z} \cdot \left(\frac{55833770631}{5000000000} - \frac{4769379582500641883561}{100000000000000000000}\right)} + \frac{313060547623}{100000000000} \cdot y\right) + x \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{y}{z} \cdot \color{blue}{\frac{-3652704169880641883561}{100000000000000000000}} + \frac{313060547623}{100000000000} \cdot y\right) + x \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{y}{z} \cdot \color{blue}{\frac{\frac{3652704169880641883561}{100000000000000000000}}{-1}} + \frac{313060547623}{100000000000} \cdot y\right) + x \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{y}{z} \cdot \frac{\color{blue}{\frac{-55833770631}{5000000000} - \frac{-4769379582500641883561}{100000000000000000000}}}{-1} + \frac{313060547623}{100000000000} \cdot y\right) + x \]

      9. times-frac N/A

         \[\leadsto \left(\color{blue}{\frac{y \cdot \left(\frac{-55833770631}{5000000000} - \frac{-4769379582500641883561}{100000000000000000000}\right)}{z \cdot -1}} + \frac{313060547623}{100000000000} \cdot y\right) + x \]

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

          \[\leadsto \left(\frac{\color{blue}{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}}{z \cdot -1} + \frac{313060547623}{100000000000} \cdot y\right) + x \]

      11. *-commutative N/A

          \[\leadsto \left(\frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{\color{blue}{-1 \cdot z}} + \frac{313060547623}{100000000000} \cdot y\right) + x \]

      12. mul-1-neg N/A

          \[\leadsto \left(\frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{\color{blue}{\mathsf{neg}\left(z\right)}} + \frac{313060547623}{100000000000} \cdot y\right) + x \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}\right)\right)} + \frac{313060547623}{100000000000} \cdot y\right) + x \]

      14. mul-1-neg N/A

          \[\leadsto \left(\color{blue}{-1 \cdot \frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z}} + \frac{313060547623}{100000000000} \cdot y\right) + x \]

      15. associate-+l+ N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \left(\frac{313060547623}{100000000000} \cdot y + x\right)} \]

   5. Simplified 86.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, -36.52704169880642, \mathsf{fma}\left(y, 3.13060547623, x\right)\right)} \]

   if -1.30000000000000006e-19 < z < 1.06e17

   1. Initial program 96.6%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + x} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot b\right) \cdot y} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{1000000000000}{607771387771} \cdot b\right)} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1000000000000}{607771387771} \cdot b, x\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{b \cdot \frac{1000000000000}{607771387771}}, x\right) \]

      6. lower-*.f64 78.0

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{b \cdot 1.6453555072203998}, x\right) \]

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(y \cdot b\right) \cdot \frac{1000000000000}{607771387771}} + x \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot b, \frac{1000000000000}{607771387771}, x\right)} \]

      3. lower-*.f64 78.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot b}, 1.6453555072203998, x\right) \]

   7. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)} \]

   if 1.06e17 < z

   1. Initial program 12.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 90.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 90.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 83.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.3e-19)
   (fma y 3.13060547623 x)
   (if (<= z 1.06e+17)
     (fma (* y b) 1.6453555072203998 x)
     (fma y 3.13060547623 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.3e-19) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.06e+17) {
		tmp = fma((y * b), 1.6453555072203998, x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.3e-19)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.06e+17)
		tmp = fma(Float64(y * b), 1.6453555072203998, x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.3e-19], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.06e+17], N[(N[(y * b), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.30000000000000006e-19 or 1.06e17 < z

   1. Initial program 16.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 88.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 88.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   if -1.30000000000000006e-19 < z < 1.06e17

   1. Initial program 96.6%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + x} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot b\right) \cdot y} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{1000000000000}{607771387771} \cdot b\right)} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1000000000000}{607771387771} \cdot b, x\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{b \cdot \frac{1000000000000}{607771387771}}, x\right) \]

      6. lower-*.f64 78.0

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{b \cdot 1.6453555072203998}, x\right) \]

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(y \cdot b\right) \cdot \frac{1000000000000}{607771387771}} + x \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot b, \frac{1000000000000}{607771387771}, x\right)} \]

      3. lower-*.f64 78.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot b}, 1.6453555072203998, x\right) \]

   7. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 83.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 1.6453555072203998, b, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.3e-19)
   (fma y 3.13060547623 x)
   (if (<= z 1.06e+17)
     (fma (* y 1.6453555072203998) b x)
     (fma y 3.13060547623 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.3e-19) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.06e+17) {
		tmp = fma((y * 1.6453555072203998), b, x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.3e-19)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.06e+17)
		tmp = fma(Float64(y * 1.6453555072203998), b, x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.3e-19], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.06e+17], N[(N[(y * 1.6453555072203998), $MachinePrecision] * b + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.30000000000000006e-19 or 1.06e17 < z

   1. Initial program 16.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 88.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 88.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   if -1.30000000000000006e-19 < z < 1.06e17

   1. Initial program 96.6%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + x} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot b\right) \cdot y} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{1000000000000}{607771387771} \cdot b\right)} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1000000000000}{607771387771} \cdot b, x\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{b \cdot \frac{1000000000000}{607771387771}}, x\right) \]

      6. lower-*.f64 78.0

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{b \cdot 1.6453555072203998}, x\right) \]

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(b \cdot \frac{1000000000000}{607771387771}\right)} + x \]

      2. lift-*.f64 N/A

         \[\leadsto y \cdot \color{blue}{\left(b \cdot \frac{1000000000000}{607771387771}\right)} + x \]

      3. *-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot b\right)} + x \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(y \cdot \frac{1000000000000}{607771387771}\right) \cdot b} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot \frac{1000000000000}{607771387771}, b, x\right)} \]

      6. lower-*.f64 78.1

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot 1.6453555072203998}, b, x\right) \]

   7. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot 1.6453555072203998, b, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 83.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.3e-19)
   (fma y 3.13060547623 x)
   (if (<= z 1.06e+17)
     (fma y (* b 1.6453555072203998) x)
     (fma y 3.13060547623 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.3e-19) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.06e+17) {
		tmp = fma(y, (b * 1.6453555072203998), x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.3e-19)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.06e+17)
		tmp = fma(y, Float64(b * 1.6453555072203998), x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.3e-19], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.06e+17], N[(y * N[(b * 1.6453555072203998), $MachinePrecision] + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.30000000000000006e-19 or 1.06e17 < z

   1. Initial program 16.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 88.2

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 88.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   if -1.30000000000000006e-19 < z < 1.06e17

   1. Initial program 96.6%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + x} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1000000000000}{607771387771} \cdot b\right) \cdot y} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{1000000000000}{607771387771} \cdot b\right)} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{1000000000000}{607771387771} \cdot b, x\right)} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{b \cdot \frac{1000000000000}{607771387771}}, x\right) \]

      6. lower-*.f64 78.0

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{b \cdot 1.6453555072203998}, x\right) \]

   5. Simplified 78.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 51.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+43}:\\ \;\;\;\;y \cdot 3.13060547623\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+91}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 3.13060547623\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -2.5e+43)
   (* y 3.13060547623)
   (if (<= y 1.25e+91) x (* y 3.13060547623))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.5e+43) {
		tmp = y * 3.13060547623;
	} else if (y <= 1.25e+91) {
		tmp = x;
	} else {
		tmp = y * 3.13060547623;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-2.5d+43)) then
        tmp = y * 3.13060547623d0
    else if (y <= 1.25d+91) then
        tmp = x
    else
        tmp = y * 3.13060547623d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -2.5e+43) {
		tmp = y * 3.13060547623;
	} else if (y <= 1.25e+91) {
		tmp = x;
	} else {
		tmp = y * 3.13060547623;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -2.5e+43:
		tmp = y * 3.13060547623
	elif y <= 1.25e+91:
		tmp = x
	else:
		tmp = y * 3.13060547623
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -2.5e+43)
		tmp = Float64(y * 3.13060547623);
	elseif (y <= 1.25e+91)
		tmp = x;
	else
		tmp = Float64(y * 3.13060547623);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -2.5e+43)
		tmp = y * 3.13060547623;
	elseif (y <= 1.25e+91)
		tmp = x;
	else
		tmp = y * 3.13060547623;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -2.5e+43], N[(y * 3.13060547623), $MachinePrecision], If[LessEqual[y, 1.25e+91], x, N[(y * 3.13060547623), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5000000000000002e43 or 1.2500000000000001e91 < y

   1. Initial program 53.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \left(\frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

      2. +-commutative N/A

         \[\leadsto y \cdot \color{blue}{\left(\frac{z \cdot \left(a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)} + \frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

      3. associate-/l* N/A

         \[\leadsto y \cdot \left(\color{blue}{z \cdot \frac{a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}} + \frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto y \cdot \color{blue}{\mathsf{fma}\left(z, \frac{a + z \cdot \left(t + z \cdot \left(\frac{55833770631}{5000000000} + \frac{313060547623}{100000000000} \cdot z\right)\right)}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}, \frac{b}{\frac{607771387771}{1000000000000} + z \cdot \left(\frac{119400905721}{10000000000} + z \cdot \left(\frac{314690115749}{10000000000} + z \cdot \left(\frac{15234687407}{1000000000} + z\right)\right)\right)}\right)} \]

   5. Simplified 52.1%

      \[\leadsto \color{blue}{y \cdot \mathsf{fma}\left(z, \frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, \frac{b}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto y \cdot \color{blue}{\frac{313060547623}{100000000000}} \]
   7. Step-by-step derivation

   8. Simplified 38.0%

      \[\leadsto y \cdot \color{blue}{3.13060547623} \]

   if -2.5000000000000002e43 < y < 1.2500000000000001e91

   1. Initial program 54.5%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

      3. lower-fma.f64 75.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   5. Simplified 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

   6. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot x}\right) \]

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

         \[\leadsto \color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot \color{blue}{\left(-1 \cdot x\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot \left(-1 \cdot x\right)} \]

      6. sub-neg N/A

         \[\leadsto \color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(-1 \cdot x\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\color{blue}{\frac{y}{x} \cdot \frac{-313060547623}{100000000000}} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot x\right) \]

      8. metadata-eval N/A

         \[\leadsto \left(\frac{y}{x} \cdot \frac{-313060547623}{100000000000} + \color{blue}{-1}\right) \cdot \left(-1 \cdot x\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x}, \frac{-313060547623}{100000000000}, -1\right)} \cdot \left(-1 \cdot x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{x}}, \frac{-313060547623}{100000000000}, -1\right) \cdot \left(-1 \cdot x\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{y}{x}, \frac{-313060547623}{100000000000}, -1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      12. lower-neg.f64 74.4

          \[\leadsto \mathsf{fma}\left(\frac{y}{x}, -3.13060547623, -1\right) \cdot \color{blue}{\left(-x\right)} \]

   8. Simplified 74.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x}, -3.13060547623, -1\right) \cdot \left(-x\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1} \cdot \left(\mathsf{neg}\left(x\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 64.5%

       \[\leadsto \color{blue}{-1} \cdot \left(-x\right) \]
   12. Step-by-step derivation

       1. distribute-rgt-neg-out N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(-1 \cdot x\right)} \]

       2. neg-mul-1 N/A

          \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

       3. remove-double-neg 64.5

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

   13. Applied egg-rr 64.5%

       \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 62.7% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, 3.13060547623, x\right) \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (fma y 3.13060547623 x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return fma(y, 3.13060547623, x);
}

Julia

function code(x, y, z, t, a, b)
	return fma(y, 3.13060547623, x)
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(y * 3.13060547623 + x), $MachinePrecision]

Derivation

1. Initial program 54.0%

   \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

   3. lower-fma.f64 64.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

5. Simplified 64.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]
6. Add Preprocessing

Alternative 18: 44.7% accurate, 79.0× speedup

Math

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

FPCore

(FPCore (x y z t a b) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

def code(x, y, z, t, a, b):
	return x

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 54.0%

   \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x + \frac{313060547623}{100000000000} \cdot y} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{313060547623}{100000000000} \cdot y + x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{y \cdot \frac{313060547623}{100000000000}} + x \]

   3. lower-fma.f64 64.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

5. Simplified 64.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right)} \]

6. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right)\right)} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot \left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot x}\right) \]

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

      \[\leadsto \color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot \left(\mathsf{neg}\left(x\right)\right)} \]

   4. mul-1-neg N/A

      \[\leadsto \left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot \color{blue}{\left(-1 \cdot x\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} - 1\right) \cdot \left(-1 \cdot x\right)} \]

   6. sub-neg N/A

      \[\leadsto \color{blue}{\left(\frac{-313060547623}{100000000000} \cdot \frac{y}{x} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(-1 \cdot x\right) \]

   7. *-commutative N/A

      \[\leadsto \left(\color{blue}{\frac{y}{x} \cdot \frac{-313060547623}{100000000000}} + \left(\mathsf{neg}\left(1\right)\right)\right) \cdot \left(-1 \cdot x\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(\frac{y}{x} \cdot \frac{-313060547623}{100000000000} + \color{blue}{-1}\right) \cdot \left(-1 \cdot x\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x}, \frac{-313060547623}{100000000000}, -1\right)} \cdot \left(-1 \cdot x\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{x}}, \frac{-313060547623}{100000000000}, -1\right) \cdot \left(-1 \cdot x\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(\frac{y}{x}, \frac{-313060547623}{100000000000}, -1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   12. lower-neg.f64 61.5

       \[\leadsto \mathsf{fma}\left(\frac{y}{x}, -3.13060547623, -1\right) \cdot \color{blue}{\left(-x\right)} \]

8. Simplified 61.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{x}, -3.13060547623, -1\right) \cdot \left(-x\right)} \]

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-1} \cdot \left(\mathsf{neg}\left(x\right)\right) \]
10. Step-by-step derivation

11. Simplified 44.5%

    \[\leadsto \color{blue}{-1} \cdot \left(-x\right) \]
12. Step-by-step derivation

    1. distribute-rgt-neg-out N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(-1 \cdot x\right)} \]

    2. neg-mul-1 N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)}\right) \]

    3. remove-double-neg 44.5

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

13. Applied egg-rr 44.5%

    \[\leadsto \color{blue}{x} \]
14. Add Preprocessing

Developer Target 1: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(\left(3.13060547623 - \frac{36.527041698806414}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{if}\;z < -6.499344996252632 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 7.066965436914287 \cdot 10^{+59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771}{\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1
         (+
          x
          (*
           (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z)))
           (/ y 1.0)))))
   (if (< z -6.499344996252632e+53)
     t_1
     (if (< z 7.066965436914287e+59)
       (+
        x
        (/
         y
         (/
          (+
           (*
            (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721)
            z)
           0.607771387771)
          (+
           (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z)
           b))))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (((3.13060547623 - (36.527041698806414 / z)) + (t / (z * z))) * (y / 1.0));
	double tmp;
	if (z < -6.499344996252632e+53) {
		tmp = t_1;
	} else if (z < 7.066965436914287e+59) {
		tmp = x + (y / ((((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771) / ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((3.13060547623d0 - (36.527041698806414d0 / z)) + (t / (z * z))) * (y / 1.0d0))
    if (z < (-6.499344996252632d+53)) then
        tmp = t_1
    else if (z < 7.066965436914287d+59) then
        tmp = x + (y / ((((((((z + 15.234687407d0) * z) + 31.4690115749d0) * z) + 11.9400905721d0) * z) + 0.607771387771d0) / ((((((((z * 3.13060547623d0) + 11.1667541262d0) * z) + t) * z) + a) * z) + b)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x + (((3.13060547623 - (36.527041698806414 / z)) + (t / (z * z))) * (y / 1.0));
	double tmp;
	if (z < -6.499344996252632e+53) {
		tmp = t_1;
	} else if (z < 7.066965436914287e+59) {
		tmp = x + (y / ((((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771) / ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x + (((3.13060547623 - (36.527041698806414 / z)) + (t / (z * z))) * (y / 1.0))
	tmp = 0
	if z < -6.499344996252632e+53:
		tmp = t_1
	elif z < 7.066965436914287e+59:
		tmp = x + (y / ((((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771) / ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x + Float64(Float64(Float64(3.13060547623 - Float64(36.527041698806414 / z)) + Float64(t / Float64(z * z))) * Float64(y / 1.0)))
	tmp = 0.0
	if (z < -6.499344996252632e+53)
		tmp = t_1;
	elseif (z < 7.066965436914287e+59)
		tmp = Float64(x + Float64(y / Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x + (((3.13060547623 - (36.527041698806414 / z)) + (t / (z * z))) * (y / 1.0));
	tmp = 0.0;
	if (z < -6.499344996252632e+53)
		tmp = t_1;
	elseif (z < 7.066965436914287e+59)
		tmp = x + (y / ((((((((z + 15.234687407) * z) + 31.4690115749) * z) + 11.9400905721) * z) + 0.607771387771) / ((((((((z * 3.13060547623) + 11.1667541262) * z) + t) * z) + a) * z) + b)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x + N[(N[(N[(3.13060547623 - N[(36.527041698806414 / z), $MachinePrecision]), $MachinePrecision] + N[(t / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -6.499344996252632e+53], t$95$1, If[Less[z, 7.066965436914287e+59], N[(x + N[(y / N[(N[(N[(N[(N[(N[(N[(N[(z + 15.234687407), $MachinePrecision] * z), $MachinePrecision] + 31.4690115749), $MachinePrecision] * z), $MachinePrecision] + 11.9400905721), $MachinePrecision] * z), $MachinePrecision] + 0.607771387771), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision] * z), $MachinePrecision] + t), $MachinePrecision] * z), $MachinePrecision] + a), $MachinePrecision] * z), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -649934499625263200000000000000000000000000000000000000) (+ x (* (+ (- 313060547623/100000000000 (/ 18263520849403207/500000000000000 z)) (/ t (* z z))) (/ y 1))) (if (< z 706696543691428700000000000000000000000000000000000000000000) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15234687407/1000000000) z) 314690115749/10000000000) z) 119400905721/10000000000) z) 607771387771/1000000000000) (+ (* (+ (* (+ (* (+ (* z 313060547623/100000000000) 55833770631/5000000000) z) t) z) a) z) b)))) (+ x (* (+ (- 313060547623/100000000000 (/ 18263520849403207/500000000000000 z)) (/ t (* z z))) (/ y 1))))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))