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

Percentage Accurate: 57.2% → 93.6%

Time: 17.5s

Alternatives: 16

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: 57.2% 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: 93.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right) + \frac{\frac{\mathsf{fma}\left(y, t, \mathsf{fma}\left(y, -98.5170599679272, y \cdot 556.47806218377\right)\right)}{z} - y \cdot 36.52704169880642}{z}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, t, a\right), b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2e+19)
   (+
    (fma y 3.13060547623 x)
    (/
     (-
      (/ (fma y t (fma y -98.5170599679272 (* y 556.47806218377))) z)
      (* y 36.52704169880642))
     z))
   (if (<= z 6e+43)
     (+
      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 <= -2e+19) {
		tmp = fma(y, 3.13060547623, x) + (((fma(y, t, fma(y, -98.5170599679272, (y * 556.47806218377))) / z) - (y * 36.52704169880642)) / z);
	} else if (z <= 6e+43) {
		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 <= -2e+19)
		tmp = Float64(fma(y, 3.13060547623, x) + Float64(Float64(Float64(fma(y, t, fma(y, -98.5170599679272, Float64(y * 556.47806218377))) / z) - Float64(y * 36.52704169880642)) / z));
	elseif (z <= 6e+43)
		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, -2e+19], N[(N[(y * 3.13060547623 + x), $MachinePrecision] + N[(N[(N[(N[(y * t + N[(y * -98.5170599679272 + N[(y * 556.47806218377), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] - N[(y * 36.52704169880642), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6e+43], 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 3 regimes

2. if z < -2e19

   1. Initial program 8.6%

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

   3. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]

   5. Applied rewrites 95.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right) - \frac{\frac{\mathsf{fma}\left(y, t, \mathsf{fma}\left(y, -98.5170599679272, y \cdot 556.47806218377\right)\right)}{-z} + y \cdot 36.52704169880642}{z}} \]

   if -2e19 < z < 6.00000000000000033e43

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 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 98.3

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

   5. Applied rewrites 98.3%

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

   if 6.00000000000000033e43 < z

   1. Initial program 7.9%

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

   3. Taylor expanded in 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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right) + \frac{\frac{\mathsf{fma}\left(y, t, \mathsf{fma}\left(y, -98.5170599679272, y \cdot 556.47806218377\right)\right)}{z} - y \cdot 36.52704169880642}{z}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, t, a\right), b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 70.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+41}:\\ \;\;\;\;b \cdot \left(y \cdot 1.6453555072203998\right)\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;y \cdot \left(b \cdot 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

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 -4e+41)
     (* b (* y 1.6453555072203998))
     (if (<= t_1 1.5e+31)
       (fma y 3.13060547623 x)
       (if (<= t_1 INFINITY)
         (* y (* b 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 <= -4e+41) {
		tmp = b * (y * 1.6453555072203998);
	} else if (t_1 <= 1.5e+31) {
		tmp = fma(y, 3.13060547623, x);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = y * (b * 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 <= -4e+41)
		tmp = Float64(b * Float64(y * 1.6453555072203998));
	elseif (t_1 <= 1.5e+31)
		tmp = fma(y, 3.13060547623, x);
	elseif (t_1 <= Inf)
		tmp = Float64(y * Float64(b * 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, -4e+41], N[(b * N[(y * 1.6453555072203998), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.5e+31], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(y * N[(b * 1.6453555072203998), $MachinePrecision]), $MachinePrecision], 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))) < -4.00000000000000002e41

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 7.5%

      \[\leadsto y \cdot 3.13060547623 \]

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 47.1%

       \[\leadsto 1.6453555072203998 \cdot \color{blue}{\left(b \cdot y\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 47.1%

       \[\leadsto \left(y \cdot 1.6453555072203998\right) \cdot b \]

   if -4.00000000000000002e41 < (/.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))) < 1.49999999999999995e31 or +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 44.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 89.3

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

   5. Applied rewrites 89.3%

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

   if 1.49999999999999995e31 < (/.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 x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 53.1%

      \[\leadsto y \cdot \left(b \cdot \color{blue}{1.6453555072203998}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq -4 \cdot 10^{+41}:\\ \;\;\;\;b \cdot \left(y \cdot 1.6453555072203998\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq 1.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;y \cdot \left(b \cdot 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+41}:\\ \;\;\;\;b \cdot \left(y \cdot 1.6453555072203998\right)\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

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 -4e+41)
     (* b (* y 1.6453555072203998))
     (if (<= t_1 1.5e+31)
       (fma y 3.13060547623 x)
       (if (<= t_1 INFINITY)
         (* 1.6453555072203998 (* y b))
         (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 <= -4e+41) {
		tmp = b * (y * 1.6453555072203998);
	} else if (t_1 <= 1.5e+31) {
		tmp = fma(y, 3.13060547623, x);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 1.6453555072203998 * (y * b);
	} 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 <= -4e+41)
		tmp = Float64(b * Float64(y * 1.6453555072203998));
	elseif (t_1 <= 1.5e+31)
		tmp = fma(y, 3.13060547623, x);
	elseif (t_1 <= Inf)
		tmp = Float64(1.6453555072203998 * Float64(y * b));
	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, -4e+41], N[(b * N[(y * 1.6453555072203998), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.5e+31], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(1.6453555072203998 * N[(y * b), $MachinePrecision]), $MachinePrecision], 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))) < -4.00000000000000002e41

   1. Initial program 88.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 7.5%

      \[\leadsto y \cdot 3.13060547623 \]

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 47.1%

       \[\leadsto 1.6453555072203998 \cdot \color{blue}{\left(b \cdot y\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 47.1%

       \[\leadsto \left(y \cdot 1.6453555072203998\right) \cdot b \]

   if -4.00000000000000002e41 < (/.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))) < 1.49999999999999995e31 or +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 44.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 89.3

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

   5. Applied rewrites 89.3%

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

   if 1.49999999999999995e31 < (/.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 x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 5.7%

      \[\leadsto y \cdot 3.13060547623 \]

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 53.0%

       \[\leadsto 1.6453555072203998 \cdot \color{blue}{\left(b \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq -4 \cdot 10^{+41}:\\ \;\;\;\;b \cdot \left(y \cdot 1.6453555072203998\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq 1.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1.6453555072203998 \cdot \left(y \cdot b\right)\\ t_2 := \frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* 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 -4e+41)
     t_1
     (if (<= t_2 1.5e+31)
       (fma y 3.13060547623 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 <= -4e+41) {
		tmp = t_1;
	} else if (t_2 <= 1.5e+31) {
		tmp = fma(y, 3.13060547623, 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 <= -4e+41)
		tmp = t_1;
	elseif (t_2 <= 1.5e+31)
		tmp = fma(y, 3.13060547623, 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, -4e+41], t$95$1, If[LessEqual[t$95$2, 1.5e+31], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, 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))) < -4.00000000000000002e41 or 1.49999999999999995e31 < (/.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 89.4%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 6.6%

      \[\leadsto y \cdot 3.13060547623 \]

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 50.1%

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

   if -4.00000000000000002e41 < (/.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))) < 1.49999999999999995e31 or +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 44.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 89.3

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

   5. Applied rewrites 89.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq -4 \cdot 10^{+41}:\\ \;\;\;\;1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq 1.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.4% accurate, 0.5× speedup

Math

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

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)
   (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 = 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 = 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[(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], 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 94.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 96.6%

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

   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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 97.2%

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

Alternative 6: 92.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right) + \frac{\frac{\mathsf{fma}\left(y, t, \mathsf{fma}\left(y, -98.5170599679272, y \cdot 556.47806218377\right)\right)}{z} - y \cdot 36.52704169880642}{z}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 11.1667541262, t\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -13.0)
   (+
    (fma y 3.13060547623 x)
    (/
     (-
      (/ (fma y t (fma y -98.5170599679272 (* y 556.47806218377))) z)
      (* y 36.52704169880642))
     z))
   (if (<= z 1.32e+42)
     (fma
      (/
       (fma z (fma z (fma z 11.1667541262 t) a) b)
       (fma
        z
        (fma z (fma z 15.234687407 31.4690115749) 11.9400905721)
        0.607771387771))
      y
      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 <= -13.0) {
		tmp = fma(y, 3.13060547623, x) + (((fma(y, t, fma(y, -98.5170599679272, (y * 556.47806218377))) / z) - (y * 36.52704169880642)) / z);
	} else if (z <= 1.32e+42) {
		tmp = fma((fma(z, fma(z, fma(z, 11.1667541262, t), a), b) / fma(z, fma(z, fma(z, 15.234687407, 31.4690115749), 11.9400905721), 0.607771387771)), y, x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -13.0)
		tmp = Float64(fma(y, 3.13060547623, x) + Float64(Float64(Float64(fma(y, t, fma(y, -98.5170599679272, Float64(y * 556.47806218377))) / z) - Float64(y * 36.52704169880642)) / z));
	elseif (z <= 1.32e+42)
		tmp = fma(Float64(fma(z, fma(z, fma(z, 11.1667541262, t), a), b) / fma(z, fma(z, fma(z, 15.234687407, 31.4690115749), 11.9400905721), 0.607771387771)), y, x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -13.0], N[(N[(y * 3.13060547623 + x), $MachinePrecision] + N[(N[(N[(N[(y * t + N[(y * -98.5170599679272 + N[(y * 556.47806218377), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] - N[(y * 36.52704169880642), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.32e+42], N[(N[(N[(z * N[(z * N[(z * 11.1667541262 + t), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision] / N[(z * N[(z * N[(z * 15.234687407 + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -13

   1. Initial program 14.3%

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

   3. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{\left(\frac{-55833770631}{5000000000} \cdot y + -1 \cdot \frac{t \cdot y - \left(\frac{-15234687407}{1000000000} \cdot \left(\frac{-55833770631}{5000000000} \cdot y - \frac{-4769379582500641883561}{100000000000000000000} \cdot y\right) + \frac{98517059967927196814627}{1000000000000000000000} \cdot y\right)}{z}\right) - \frac{-4769379582500641883561}{100000000000000000000} \cdot y}{z} + \frac{313060547623}{100000000000} \cdot y\right)} \]

   5. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right) - \frac{\frac{\mathsf{fma}\left(y, t, \mathsf{fma}\left(y, -98.5170599679272, y \cdot 556.47806218377\right)\right)}{-z} + y \cdot 36.52704169880642}{z}} \]

   if -13 < z < 1.32e42

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 94.9

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

   5. Applied rewrites 94.9%

      \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(z, a, b\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.7

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

   10. Applied rewrites 93.7%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 97.5

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

   13. Applied rewrites 97.5%

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

   if 1.32e42 < z

   1. Initial program 7.9%

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

   3. Taylor expanded in 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 98.0

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

   5. Applied rewrites 98.0%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right) + \frac{\frac{\mathsf{fma}\left(y, t, \mathsf{fma}\left(y, -98.5170599679272, y \cdot 556.47806218377\right)\right)}{z} - y \cdot 36.52704169880642}{z}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 11.1667541262, t\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 11.1667541262, t\right), a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -13.0)
   (fma y 3.13060547623 x)
   (if (<= z 1.32e+42)
     (fma
      (/
       (fma z (fma z (fma z 11.1667541262 t) a) b)
       (fma
        z
        (fma z (fma z 15.234687407 31.4690115749) 11.9400905721)
        0.607771387771))
      y
      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 <= -13.0) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.32e+42) {
		tmp = fma((fma(z, fma(z, fma(z, 11.1667541262, t), a), b) / fma(z, fma(z, fma(z, 15.234687407, 31.4690115749), 11.9400905721), 0.607771387771)), y, x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -13.0)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.32e+42)
		tmp = fma(Float64(fma(z, fma(z, fma(z, 11.1667541262, t), a), b) / fma(z, fma(z, fma(z, 15.234687407, 31.4690115749), 11.9400905721), 0.607771387771)), y, x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -13.0], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.32e+42], N[(N[(N[(z * N[(z * N[(z * 11.1667541262 + t), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision] / N[(z * N[(z * N[(z * 15.234687407 + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -13 or 1.32e42 < z

   1. Initial program 11.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 94.8

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

   5. Applied rewrites 94.8%

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

   if -13 < z < 1.32e42

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 94.9

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

   5. Applied rewrites 94.9%

      \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(z, a, b\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.7

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

   10. Applied rewrites 93.7%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 97.5

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

   13. Applied rewrites 97.5%

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

Alternative 8: 92.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, t, a\right), b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -13.0)
   (fma y 3.13060547623 x)
   (if (<= z 1.32e+42)
     (fma
      (/
       (fma z (fma z t a) b)
       (fma
        z
        (fma z (fma z 15.234687407 31.4690115749) 11.9400905721)
        0.607771387771))
      y
      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 <= -13.0) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.32e+42) {
		tmp = fma((fma(z, fma(z, t, a), b) / fma(z, fma(z, fma(z, 15.234687407, 31.4690115749), 11.9400905721), 0.607771387771)), y, x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -13.0)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.32e+42)
		tmp = fma(Float64(fma(z, fma(z, t, a), b) / fma(z, fma(z, fma(z, 15.234687407, 31.4690115749), 11.9400905721), 0.607771387771)), y, x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -13.0], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.32e+42], N[(N[(N[(z * N[(z * t + a), $MachinePrecision] + b), $MachinePrecision] / N[(z * N[(z * N[(z * 15.234687407 + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -13 or 1.32e42 < z

   1. Initial program 11.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 94.8

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

   5. Applied rewrites 94.8%

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

   if -13 < z < 1.32e42

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 94.9

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

   5. Applied rewrites 94.9%

      \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(z, a, b\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.7

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

   10. Applied rewrites 93.7%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 97.1

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

   13. Applied rewrites 97.1%

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

Alternative 9: 90.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, a, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2e+19)
   (fma y 3.13060547623 x)
   (if (<= z 9.5e+38)
     (fma
      (/
       (fma z a b)
       (fma
        z
        (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
        0.607771387771))
      y
      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 <= -2e+19) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 9.5e+38) {
		tmp = fma((fma(z, a, b) / fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)), y, x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2e+19)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 9.5e+38)
		tmp = fma(Float64(fma(z, a, b) / fma(z, fma(z, fma(z, Float64(z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)), y, x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2e+19], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 9.5e+38], N[(N[(N[(z * a + b), $MachinePrecision] / N[(z * N[(z * N[(z * N[(z + 15.234687407), $MachinePrecision] + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e19 or 9.4999999999999995e38 < z

   1. Initial program 8.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   if -2e19 < z < 9.4999999999999995e38

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 95.2

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

   5. Applied rewrites 95.2%

      \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(z, a, b\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

   7. Applied rewrites 95.2%

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

Alternative 10: 89.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+36}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(z, a, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 31.4690115749, 11.9400905721\right), 0.607771387771\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -9e+17)
   (fma y 3.13060547623 x)
   (if (<= z 1.25e+36)
     (+
      x
      (/
       (* y (fma z a b))
       (fma z (fma z 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 <= -9e+17) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.25e+36) {
		tmp = x + ((y * fma(z, a, b)) / fma(z, fma(z, 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 <= -9e+17)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.25e+36)
		tmp = Float64(x + Float64(Float64(y * fma(z, a, b)) / fma(z, fma(z, 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, -9e+17], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.25e+36], N[(x + N[(N[(y * N[(z * a + b), $MachinePrecision]), $MachinePrecision] / N[(z * N[(z * 31.4690115749 + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9e17 or 1.24999999999999994e36 < z

   1. Initial program 8.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   if -9e17 < z < 1.24999999999999994e36

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.6

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

   8. Applied rewrites 93.6%

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

Alternative 11: 89.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, a, b\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 31.4690115749, 11.9400905721\right), 0.607771387771\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -9e+17)
   (fma y 3.13060547623 x)
   (if (<= z 1.25e+36)
     (fma
      (/
       (fma z a b)
       (fma z (fma z 31.4690115749 11.9400905721) 0.607771387771))
      y
      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 <= -9e+17) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.25e+36) {
		tmp = fma((fma(z, a, b) / fma(z, fma(z, 31.4690115749, 11.9400905721), 0.607771387771)), y, x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -9e+17)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.25e+36)
		tmp = fma(Float64(fma(z, a, b) / fma(z, fma(z, 31.4690115749, 11.9400905721), 0.607771387771)), y, x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -9e+17], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.25e+36], N[(N[(N[(z * a + b), $MachinePrecision] / N[(z * N[(z * 31.4690115749 + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9e17 or 1.24999999999999994e36 < z

   1. Initial program 8.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   if -9e17 < z < 1.24999999999999994e36

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 95.2

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

   5. Applied rewrites 95.2%

      \[\leadsto x + \frac{y \cdot \color{blue}{\mathsf{fma}\left(z, a, b\right)}}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

   7. Applied rewrites 95.2%

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

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(z, a, b\right)}{\mathsf{fma}\left(z, \color{blue}{\frac{119400905721}{10000000000} + \frac{314690115749}{10000000000} \cdot z}, \frac{607771387771}{1000000000000}\right)}, y, x\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 93.6

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

   10. Applied rewrites 93.6%

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

Alternative 12: 89.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+30}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(z, a, b\right)}{0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -9e+17)
   (fma y 3.13060547623 x)
   (if (<= z 1.2e+30)
     (+ x (/ (* y (fma z a b)) 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 <= -9e+17) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.2e+30) {
		tmp = x + ((y * fma(z, a, b)) / 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 <= -9e+17)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.2e+30)
		tmp = Float64(x + Float64(Float64(y * fma(z, a, b)) / 0.607771387771));
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -9e+17], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.2e+30], N[(x + N[(N[(y * N[(z * a + b), $MachinePrecision]), $MachinePrecision] / 0.607771387771), $MachinePrecision]), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -9e17 or 1.2e30 < z

   1. Initial program 8.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   if -9e17 < z < 1.2e30

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 95.2

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

   5. Applied rewrites 95.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 92.8%

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

Alternative 13: 82.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, \mathsf{fma}\left(\frac{y}{z}, -36.52704169880642, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.2e+17)
   (fma y 3.13060547623 x)
   (if (<= z 5.3e-9)
     (fma y (* b 1.6453555072203998) x)
     (fma y 3.13060547623 (fma (/ y z) -36.52704169880642 x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.2e+17) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 5.3e-9) {
		tmp = fma(y, (b * 1.6453555072203998), x);
	} else {
		tmp = fma(y, 3.13060547623, fma((y / z), -36.52704169880642, x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.2e+17)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 5.3e-9)
		tmp = fma(y, Float64(b * 1.6453555072203998), x);
	else
		tmp = fma(y, 3.13060547623, fma(Float64(y / z), -36.52704169880642, x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.2e+17], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 5.3e-9], N[(y * N[(b * 1.6453555072203998), $MachinePrecision] + x), $MachinePrecision], N[(y * 3.13060547623 + N[(N[(y / z), $MachinePrecision] * -36.52704169880642 + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e17

   1. Initial program 10.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 93.4

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

   5. Applied rewrites 93.4%

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

   if -2.2e17 < z < 5.30000000000000031e-9

   1. Initial program 99.7%

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

   3. Taylor expanded in 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 82.1

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

   5. Applied rewrites 82.1%

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

   if 5.30000000000000031e-9 < z

   1. Initial program 15.7%

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

   3. Taylor expanded in 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. associate--l+ N/A

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

      3. +-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. times-frac N/A

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

      9. *-commutative N/A

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. mul-1-neg N/A

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

      14. associate-+r+ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 92.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, \mathsf{fma}\left(\frac{y}{z}, -36.52704169880642, x\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 82.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.2e+17)
   (fma y 3.13060547623 x)
   (if (<= z 1.55e-6)
     (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 <= -2.2e+17) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.55e-6) {
		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 <= -2.2e+17)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.55e-6)
		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, -2.2e+17], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.55e-6], 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 < -2.2e17 or 1.55e-6 < z

   1. Initial program 12.1%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 93.5

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

   5. Applied rewrites 93.5%

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

   if -2.2e17 < z < 1.55e-6

   1. Initial program 99.7%

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

   3. Taylor expanded in 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 81.6

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

   5. Applied rewrites 81.6%

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

Alternative 15: 62.9% 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 59.0%

   \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
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 68.5

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

5. Applied rewrites 68.5%

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

Alternative 16: 22.7% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ y \cdot 3.13060547623 \end{array} \]

FPCore

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

C

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

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 = y * 3.13060547623d0
end function

Java

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

Python

def code(x, y, z, t, a, b):
	return y * 3.13060547623

Julia

function code(x, y, z, t, a, b)
	return Float64(y * 3.13060547623)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = y * 3.13060547623;
end

Wolfram

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

Derivation

1. Initial program 59.0%

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

3. Taylor expanded in x around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

5. Applied rewrites 32.4%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 22.3%

   \[\leadsto y \cdot 3.13060547623 \]
9. Add Preprocessing

Developer Target 1: 98.4% 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 2024233 
(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))))