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

Percentage Accurate: 58.8% → 98.1%

Time: 18.2s

Alternatives: 18

Speedup: 11.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 58.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- 3.13060547623 (/ t (* z (- z)))) x)))
   (if (<= z -3.8e+50)
     t_1
     (if (<= z 4.2e+32)
       (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)
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (3.13060547623 - (t / (z * -z))), x);
	double tmp;
	if (z <= -3.8e+50) {
		tmp = t_1;
	} else if (z <= 4.2e+32) {
		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 = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(3.13060547623 - Float64(t / Float64(z * Float64(-z)))), x)
	tmp = 0.0
	if (z <= -3.8e+50)
		tmp = t_1;
	elseif (z <= 4.2e+32)
		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 = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(3.13060547623 - N[(t / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -3.8e+50], t$95$1, If[LessEqual[z, 4.2e+32], 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], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.79999999999999987e50 or 4.2000000000000001e32 < z

   1. Initial program 12.7%

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

   3. Taylor expanded in z around -inf

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - -1 \cdot \frac{t}{\color{blue}{{z}^{2}}}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 99.9%

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

   if -3.79999999999999987e50 < z < 4.2000000000000001e32

   1. Initial program 97.7%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   2. Add Preprocessing
   3. 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 99.0%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.1% accurate, 0.5× speedup

Math

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

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)
   (+
    x
    (/
     y
     (/
      (fma
       z
       (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
       0.607771387771)
      (fma z (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a) b))))
   (fma y (- 3.13060547623 (/ t (* z (- z)))) 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 = x + (y / (fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771) / fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b)));
	} else {
		tmp = fma(y, (3.13060547623 - (t / (z * -z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771)) <= Inf)
		tmp = Float64(x + Float64(y / Float64(fma(z, fma(z, fma(z, Float64(z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771) / fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b))));
	else
		tmp = fma(y, Float64(3.13060547623 - Float64(t / Float64(z * Float64(-z)))), 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[(x + N[(y / N[(N[(z * N[(z * N[(z * N[(z + 15.234687407), $MachinePrecision] + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision] / N[(z * N[(z * N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(3.13060547623 - N[(t / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}{\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}} \]

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.9

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

   4. Applied rewrites 97.9%

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

   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

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - -1 \cdot \frac{t}{\color{blue}{{z}^{2}}}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 99.9%

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

4. Final simplification 98.6%

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

Alternative 3: 66.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq -4 \cdot 10^{+88}:\\ \;\;\;\;\left(y \cdot b\right) \cdot 1.6453555072203998\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

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))
      -4e+88)
   (* (* 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 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)) <= -4e+88) {
		tmp = (y * b) * 1.6453555072203998;
	} 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)) <= -4e+88)
		tmp = Float64(Float64(y * b) * 1.6453555072203998);
	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], -4e+88], N[(N[(y * b), $MachinePrecision] * 1.6453555072203998), $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))) < -3.99999999999999984e88

   1. Initial program 89.2%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 64.9

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

   5. Applied rewrites 64.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 51.2%

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

   if -3.99999999999999984e88 < (/.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 58.5%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 70.3

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

   5. Applied rewrites 70.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq -4 \cdot 10^{+88}:\\ \;\;\;\;\left(y \cdot b\right) \cdot 1.6453555072203998\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-22}:\\ \;\;\;\;x + \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} \cdot \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, t, a\right), y \cdot b\right)\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \frac{y}{\frac{0.607771387771}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \frac{\frac{t + 457.9610022158428}{z} - 36.52704169880642}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.8e+33)
   (fma y (- 3.13060547623 (/ t (* z (- z)))) x)
   (if (<= z -1.02e-22)
     (+
      x
      (*
       (/
        1.0
        (fma
         z
         (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
         0.607771387771))
       (fma z (* y (fma z t a)) (* y b))))
     (if (<= z 190.0)
       (+
        x
        (/
         y
         (/
          0.607771387771
          (fma z (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a) b))))
       (fma
        (+
         3.13060547623
         (/ (- (/ (+ t 457.9610022158428) z) 36.52704169880642) z))
        y
        x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.8e+33) {
		tmp = fma(y, (3.13060547623 - (t / (z * -z))), x);
	} else if (z <= -1.02e-22) {
		tmp = x + ((1.0 / fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)) * fma(z, (y * fma(z, t, a)), (y * b)));
	} else if (z <= 190.0) {
		tmp = x + (y / (0.607771387771 / fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b)));
	} else {
		tmp = fma((3.13060547623 + ((((t + 457.9610022158428) / z) - 36.52704169880642) / z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.8e+33)
		tmp = fma(y, Float64(3.13060547623 - Float64(t / Float64(z * Float64(-z)))), x);
	elseif (z <= -1.02e-22)
		tmp = Float64(x + Float64(Float64(1.0 / fma(z, fma(z, fma(z, Float64(z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)) * fma(z, Float64(y * fma(z, t, a)), Float64(y * b))));
	elseif (z <= 190.0)
		tmp = Float64(x + Float64(y / Float64(0.607771387771 / fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b))));
	else
		tmp = fma(Float64(3.13060547623 + Float64(Float64(Float64(Float64(t + 457.9610022158428) / z) - 36.52704169880642) / z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.8e+33], N[(y * N[(3.13060547623 - N[(t / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, -1.02e-22], N[(x + N[(N[(1.0 / N[(z * N[(z * N[(z * N[(z + 15.234687407), $MachinePrecision] + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision] * N[(z * N[(y * N[(z * t + a), $MachinePrecision]), $MachinePrecision] + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 190.0], N[(x + N[(y / N[(0.607771387771 / N[(z * N[(z * N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.13060547623 + N[(N[(N[(N[(t + 457.9610022158428), $MachinePrecision] / z), $MachinePrecision] - 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.8000000000000001e33

   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

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - -1 \cdot \frac{t}{\color{blue}{{z}^{2}}}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 99.9%

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

   if -1.8000000000000001e33 < z < -1.02000000000000002e-22

   1. Initial program 93.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right) \cdot \frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      3. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

   7. Applied rewrites 91.7%

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

   if -1.02000000000000002e-22 < z < 190

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}{\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}} \]

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 99.7%

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

   if 190 < z

   1. Initial program 29.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 85.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.5%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.6%

       \[\leadsto \mathsf{fma}\left(3.13060547623 - \frac{36.52704169880642 - \frac{t + 457.9610022158428}{z}}{z}, y, x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-22}:\\ \;\;\;\;x + \frac{1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)} \cdot \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, t, a\right), y \cdot b\right)\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \frac{y}{\frac{0.607771387771}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \frac{\frac{t + 457.9610022158428}{z} - 36.52704169880642}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z \cdot t\_1}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \frac{y}{\frac{0.607771387771}{\mathsf{fma}\left(z, t\_1, b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \frac{\frac{t + 457.9610022158428}{z} - 36.52704169880642}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a)))
   (if (<= z -3.8e+50)
     (fma y (- 3.13060547623 (/ t (* z (- z)))) x)
     (if (<= z -1.3e-10)
       (fma
        y
        (/
         (* z t_1)
         (fma
          z
          (fma z (fma z (+ z 15.234687407) 31.4690115749) 11.9400905721)
          0.607771387771))
        x)
       (if (<= z 190.0)
         (+ x (/ y (/ 0.607771387771 (fma z t_1 b))))
         (fma
          (+
           3.13060547623
           (/ (- (/ (+ t 457.9610022158428) z) 36.52704169880642) z))
          y
          x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a);
	double tmp;
	if (z <= -3.8e+50) {
		tmp = fma(y, (3.13060547623 - (t / (z * -z))), x);
	} else if (z <= -1.3e-10) {
		tmp = fma(y, ((z * t_1) / fma(z, fma(z, fma(z, (z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)), x);
	} else if (z <= 190.0) {
		tmp = x + (y / (0.607771387771 / fma(z, t_1, b)));
	} else {
		tmp = fma((3.13060547623 + ((((t + 457.9610022158428) / z) - 36.52704169880642) / z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a)
	tmp = 0.0
	if (z <= -3.8e+50)
		tmp = fma(y, Float64(3.13060547623 - Float64(t / Float64(z * Float64(-z)))), x);
	elseif (z <= -1.3e-10)
		tmp = fma(y, Float64(Float64(z * t_1) / fma(z, fma(z, fma(z, Float64(z + 15.234687407), 31.4690115749), 11.9400905721), 0.607771387771)), x);
	elseif (z <= 190.0)
		tmp = Float64(x + Float64(y / Float64(0.607771387771 / fma(z, t_1, b))));
	else
		tmp = fma(Float64(3.13060547623 + Float64(Float64(Float64(Float64(t + 457.9610022158428) / z) - 36.52704169880642) / z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[z, -3.8e+50], N[(y * N[(3.13060547623 - N[(t / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, -1.3e-10], N[(y * N[(N[(z * t$95$1), $MachinePrecision] / N[(z * N[(z * N[(z * N[(z + 15.234687407), $MachinePrecision] + 31.4690115749), $MachinePrecision] + 11.9400905721), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 190.0], N[(x + N[(y / N[(0.607771387771 / N[(z * t$95$1 + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.13060547623 + N[(N[(N[(N[(t + 457.9610022158428), $MachinePrecision] / z), $MachinePrecision] - 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.79999999999999987e50

   1. Initial program 7.4%

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

   3. Taylor expanded in z around -inf

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - -1 \cdot \frac{t}{\color{blue}{{z}^{2}}}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 99.9%

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

   if -3.79999999999999987e50 < z < -1.29999999999999991e-10

   1. Initial program 89.8%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 89.3%

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

   if -1.29999999999999991e-10 < z < 190

   1. Initial program 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}{\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}} \]

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 99.6%

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

   if 190 < z

   1. Initial program 29.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 85.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.5%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.6%

       \[\leadsto \mathsf{fma}\left(3.13060547623 - \frac{36.52704169880642 - \frac{t + 457.9610022158428}{z}}{z}, y, x\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq -1.3 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, z + 15.234687407, 31.4690115749\right), 11.9400905721\right), 0.607771387771\right)}, x\right)\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \frac{y}{\frac{0.607771387771}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \frac{\frac{t + 457.9610022158428}{z} - 36.52704169880642}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \frac{y}{\frac{0.607771387771}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \frac{\frac{t + 457.9610022158428}{z} - 36.52704169880642}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.6e+33)
   (fma y (- 3.13060547623 (/ t (* z (- z)))) x)
   (if (<= z 190.0)
     (+
      x
      (/
       y
       (/
        0.607771387771
        (fma z (fma z (fma z (fma z 3.13060547623 11.1667541262) t) a) b))))
     (fma
      (+
       3.13060547623
       (/ (- (/ (+ t 457.9610022158428) z) 36.52704169880642) z))
      y
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.6e+33) {
		tmp = fma(y, (3.13060547623 - (t / (z * -z))), x);
	} else if (z <= 190.0) {
		tmp = x + (y / (0.607771387771 / fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b)));
	} else {
		tmp = fma((3.13060547623 + ((((t + 457.9610022158428) / z) - 36.52704169880642) / z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.6e+33)
		tmp = fma(y, Float64(3.13060547623 - Float64(t / Float64(z * Float64(-z)))), x);
	elseif (z <= 190.0)
		tmp = Float64(x + Float64(y / Float64(0.607771387771 / fma(z, fma(z, fma(z, fma(z, 3.13060547623, 11.1667541262), t), a), b))));
	else
		tmp = fma(Float64(3.13060547623 + Float64(Float64(Float64(Float64(t + 457.9610022158428) / z) - 36.52704169880642) / z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.6e+33], N[(y * N[(3.13060547623 - N[(t / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 190.0], N[(x + N[(y / N[(0.607771387771 / N[(z * N[(z * N[(z * N[(z * 3.13060547623 + 11.1667541262), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.13060547623 + N[(N[(N[(N[(t + 457.9610022158428), $MachinePrecision] / z), $MachinePrecision] - 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.60000000000000009e33

   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

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - -1 \cdot \frac{t}{\color{blue}{{z}^{2}}}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 99.9%

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

   if -1.60000000000000009e33 < z < 190

   1. Initial program 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

         \[\leadsto x + y \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}{\left(\left(\left(z \cdot \frac{313060547623}{100000000000} + \frac{55833770631}{5000000000}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}} \]

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 94.9%

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

   if 190 < z

   1. Initial program 29.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 85.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.5%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.6%

       \[\leadsto \mathsf{fma}\left(3.13060547623 - \frac{36.52704169880642 - \frac{t + 457.9610022158428}{z}}{z}, y, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \frac{y}{\frac{0.607771387771}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \frac{\frac{t + 457.9610022158428}{z} - 36.52704169880642}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, t, a\right), y \cdot b\right) \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 549.8376187179895, -32.324150453290734\right), 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \frac{\frac{t + 457.9610022158428}{z} - 36.52704169880642}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.6e+33)
   (fma y (- 3.13060547623 (/ t (* z (- z)))) x)
   (if (<= z 190.0)
     (+
      x
      (*
       (fma z (* y (fma z t a)) (* y b))
       (fma
        z
        (fma z 549.8376187179895 -32.324150453290734)
        1.6453555072203998)))
     (fma
      (+
       3.13060547623
       (/ (- (/ (+ t 457.9610022158428) z) 36.52704169880642) z))
      y
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.6e+33) {
		tmp = fma(y, (3.13060547623 - (t / (z * -z))), x);
	} else if (z <= 190.0) {
		tmp = x + (fma(z, (y * fma(z, t, a)), (y * b)) * fma(z, fma(z, 549.8376187179895, -32.324150453290734), 1.6453555072203998));
	} else {
		tmp = fma((3.13060547623 + ((((t + 457.9610022158428) / z) - 36.52704169880642) / z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.6e+33)
		tmp = fma(y, Float64(3.13060547623 - Float64(t / Float64(z * Float64(-z)))), x);
	elseif (z <= 190.0)
		tmp = Float64(x + Float64(fma(z, Float64(y * fma(z, t, a)), Float64(y * b)) * fma(z, fma(z, 549.8376187179895, -32.324150453290734), 1.6453555072203998)));
	else
		tmp = fma(Float64(3.13060547623 + Float64(Float64(Float64(Float64(t + 457.9610022158428) / z) - 36.52704169880642) / z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.6e+33], N[(y * N[(3.13060547623 - N[(t / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 190.0], N[(x + N[(N[(z * N[(y * N[(z * t + a), $MachinePrecision]), $MachinePrecision] + N[(y * b), $MachinePrecision]), $MachinePrecision] * N[(z * N[(z * 549.8376187179895 + -32.324150453290734), $MachinePrecision] + 1.6453555072203998), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.13060547623 + N[(N[(N[(N[(t + 457.9610022158428), $MachinePrecision] / z), $MachinePrecision] - 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.60000000000000009e33

   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

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - -1 \cdot \frac{t}{\color{blue}{{z}^{2}}}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 99.9%

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

   if -1.60000000000000009e33 < z < 190

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right) \cdot \frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      3. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

   7. Applied rewrites 93.4%

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

   8. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\left(\frac{1000000000000}{607771387771} + z \cdot \left(\frac{123439798033292669987862100000000000000}{224502278183706222041215714334315011} \cdot z - \frac{11940090572100000000000000}{369386059793087248348441}\right)\right)} \cdot \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, t, a\right), y \cdot b\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 90.5

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

   10. Applied rewrites 90.5%

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

   if 190 < z

   1. Initial program 29.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 85.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.5%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.6%

       \[\leadsto \mathsf{fma}\left(3.13060547623 - \frac{36.52704169880642 - \frac{t + 457.9610022158428}{z}}{z}, y, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, t, a\right), y \cdot b\right) \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, 549.8376187179895, -32.324150453290734\right), 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \frac{\frac{t + 457.9610022158428}{z} - 36.52704169880642}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq 0.051:\\ \;\;\;\;x + \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, t, a\right), y \cdot b\right) \cdot \mathsf{fma}\left(z, -32.324150453290734, 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \frac{\frac{t + 457.9610022158428}{z} - 36.52704169880642}{z}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.6e+33)
   (fma y (- 3.13060547623 (/ t (* z (- z)))) x)
   (if (<= z 0.051)
     (+
      x
      (*
       (fma z (* y (fma z t a)) (* y b))
       (fma z -32.324150453290734 1.6453555072203998)))
     (fma
      (+
       3.13060547623
       (/ (- (/ (+ t 457.9610022158428) z) 36.52704169880642) z))
      y
      x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.6e+33) {
		tmp = fma(y, (3.13060547623 - (t / (z * -z))), x);
	} else if (z <= 0.051) {
		tmp = x + (fma(z, (y * fma(z, t, a)), (y * b)) * fma(z, -32.324150453290734, 1.6453555072203998));
	} else {
		tmp = fma((3.13060547623 + ((((t + 457.9610022158428) / z) - 36.52704169880642) / z)), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.6e+33)
		tmp = fma(y, Float64(3.13060547623 - Float64(t / Float64(z * Float64(-z)))), x);
	elseif (z <= 0.051)
		tmp = Float64(x + Float64(fma(z, Float64(y * fma(z, t, a)), Float64(y * b)) * fma(z, -32.324150453290734, 1.6453555072203998)));
	else
		tmp = fma(Float64(3.13060547623 + Float64(Float64(Float64(Float64(t + 457.9610022158428) / z) - 36.52704169880642) / z)), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.6e+33], N[(y * N[(3.13060547623 - N[(t / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 0.051], N[(x + N[(N[(z * N[(y * N[(z * t + a), $MachinePrecision]), $MachinePrecision] + N[(y * b), $MachinePrecision]), $MachinePrecision] * N[(z * -32.324150453290734 + 1.6453555072203998), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.13060547623 + N[(N[(N[(N[(t + 457.9610022158428), $MachinePrecision] / z), $MachinePrecision] - 36.52704169880642), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.60000000000000009e33

   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

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - -1 \cdot \frac{t}{\color{blue}{{z}^{2}}}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 99.9%

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

   if -1.60000000000000009e33 < z < 0.0509999999999999967

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right) \cdot \frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      3. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

   7. Applied rewrites 93.4%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 90.4

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

   10. Applied rewrites 90.4%

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

   if 0.0509999999999999967 < z

   1. Initial program 29.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 85.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 95.5%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.6%

       \[\leadsto \mathsf{fma}\left(3.13060547623 - \frac{36.52704169880642 - \frac{t + 457.9610022158428}{z}}{z}, y, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq 0.051:\\ \;\;\;\;x + \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, t, a\right), y \cdot b\right) \cdot \mathsf{fma}\left(z, -32.324150453290734, 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.13060547623 + \frac{\frac{t + 457.9610022158428}{z} - 36.52704169880642}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.051:\\ \;\;\;\;x + \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, t, a\right), y \cdot b\right) \cdot \mathsf{fma}\left(z, -32.324150453290734, 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- 3.13060547623 (/ t (* z (- z)))) x)))
   (if (<= z -1.6e+33)
     t_1
     (if (<= z 0.051)
       (+
        x
        (*
         (fma z (* y (fma z t a)) (* y b))
         (fma z -32.324150453290734 1.6453555072203998)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (3.13060547623 - (t / (z * -z))), x);
	double tmp;
	if (z <= -1.6e+33) {
		tmp = t_1;
	} else if (z <= 0.051) {
		tmp = x + (fma(z, (y * fma(z, t, a)), (y * b)) * fma(z, -32.324150453290734, 1.6453555072203998));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(3.13060547623 - Float64(t / Float64(z * Float64(-z)))), x)
	tmp = 0.0
	if (z <= -1.6e+33)
		tmp = t_1;
	elseif (z <= 0.051)
		tmp = Float64(x + Float64(fma(z, Float64(y * fma(z, t, a)), Float64(y * b)) * fma(z, -32.324150453290734, 1.6453555072203998)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(3.13060547623 - N[(t / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.6e+33], t$95$1, If[LessEqual[z, 0.051], N[(x + N[(N[(z * N[(y * N[(z * t + a), $MachinePrecision]), $MachinePrecision] + N[(y * b), $MachinePrecision]), $MachinePrecision] * N[(z * -32.324150453290734 + 1.6453555072203998), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000009e33 or 0.0509999999999999967 < z

   1. Initial program 20.7%

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

   3. Taylor expanded in z around -inf

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.7%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - -1 \cdot \frac{t}{\color{blue}{{z}^{2}}}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 97.7%

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

   if -1.60000000000000009e33 < z < 0.0509999999999999967

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right) \cdot \frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      3. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

   7. Applied rewrites 93.4%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 90.4

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

   10. Applied rewrites 90.4%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq 0.051:\\ \;\;\;\;x + \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, t, a\right), y \cdot b\right) \cdot \mathsf{fma}\left(z, -32.324150453290734, 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;x + 1.6453555072203998 \cdot \left(\mathsf{fma}\left(z, t, a\right) \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- 3.13060547623 (/ t (* z (- z)))) x)))
   (if (<= z -2.5e+31)
     t_1
     (if (<= z -1.2e-71)
       (+ x (* 1.6453555072203998 (* (fma z t a) (* y z))))
       (if (<= z 2.1e-11) (fma (* y b) 1.6453555072203998 x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (3.13060547623 - (t / (z * -z))), x);
	double tmp;
	if (z <= -2.5e+31) {
		tmp = t_1;
	} else if (z <= -1.2e-71) {
		tmp = x + (1.6453555072203998 * (fma(z, t, a) * (y * z)));
	} else if (z <= 2.1e-11) {
		tmp = fma((y * b), 1.6453555072203998, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(3.13060547623 - Float64(t / Float64(z * Float64(-z)))), x)
	tmp = 0.0
	if (z <= -2.5e+31)
		tmp = t_1;
	elseif (z <= -1.2e-71)
		tmp = Float64(x + Float64(1.6453555072203998 * Float64(fma(z, t, a) * Float64(y * z))));
	elseif (z <= 2.1e-11)
		tmp = fma(Float64(y * b), 1.6453555072203998, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(3.13060547623 - N[(t / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2.5e+31], t$95$1, If[LessEqual[z, -1.2e-71], N[(x + N[(1.6453555072203998 * N[(N[(z * t + a), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e-11], N[(N[(y * b), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.50000000000000013e31 or 2.0999999999999999e-11 < z

   1. Initial program 21.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 86.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.9%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - -1 \cdot \frac{t}{\color{blue}{{z}^{2}}}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 96.9%

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

   if -2.50000000000000013e31 < z < -1.2e-71

   1. Initial program 96.5%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right) \cdot \frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      3. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 80.2%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 69.6%

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

   if -1.2e-71 < z < 2.0999999999999999e-11

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 43.6

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

   5. Applied rewrites 43.6%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 86.9

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

   8. Applied rewrites 86.9%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;x + 1.6453555072203998 \cdot \left(\mathsf{fma}\left(z, t, a\right) \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 90.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{if}\;z \leq -1.6 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, t, a\right), y \cdot b\right) \cdot 1.6453555072203998\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- 3.13060547623 (/ t (* z (- z)))) x)))
   (if (<= z -1.6e+33)
     t_1
     (if (<= z 190.0)
       (+ x (* (fma z (* y (fma z t a)) (* y b)) 1.6453555072203998))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (3.13060547623 - (t / (z * -z))), x);
	double tmp;
	if (z <= -1.6e+33) {
		tmp = t_1;
	} else if (z <= 190.0) {
		tmp = x + (fma(z, (y * fma(z, t, a)), (y * b)) * 1.6453555072203998);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(3.13060547623 - Float64(t / Float64(z * Float64(-z)))), x)
	tmp = 0.0
	if (z <= -1.6e+33)
		tmp = t_1;
	elseif (z <= 190.0)
		tmp = Float64(x + Float64(fma(z, Float64(y * fma(z, t, a)), Float64(y * b)) * 1.6453555072203998));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(3.13060547623 - N[(t / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.6e+33], t$95$1, If[LessEqual[z, 190.0], N[(x + N[(N[(z * N[(y * N[(z * t + a), $MachinePrecision]), $MachinePrecision] + N[(y * b), $MachinePrecision]), $MachinePrecision] * 1.6453555072203998), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.60000000000000009e33 or 190 < z

   1. Initial program 20.7%

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

   3. Taylor expanded in z around -inf

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.7%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - -1 \cdot \frac{t}{\color{blue}{{z}^{2}}}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 97.7%

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

   if -1.60000000000000009e33 < z < 190

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right) \cdot \frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      3. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

   7. Applied rewrites 93.4%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 90.0%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq 190:\\ \;\;\;\;x + \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, t, a\right), y \cdot b\right) \cdot 1.6453555072203998\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 89.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(z, 1.6453555072203998 \cdot \left(y \cdot a\right), \mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- 3.13060547623 (/ t (* z (- z)))) x)))
   (if (<= z -4.2e-8)
     t_1
     (if (<= z 4.6e-5)
       (fma
        z
        (* 1.6453555072203998 (* y a))
        (fma y (* b 1.6453555072203998) x))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, (3.13060547623 - (t / (z * -z))), x);
	double tmp;
	if (z <= -4.2e-8) {
		tmp = t_1;
	} else if (z <= 4.6e-5) {
		tmp = fma(z, (1.6453555072203998 * (y * a)), fma(y, (b * 1.6453555072203998), x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(3.13060547623 - Float64(t / Float64(z * Float64(-z)))), x)
	tmp = 0.0
	if (z <= -4.2e-8)
		tmp = t_1;
	elseif (z <= 4.6e-5)
		tmp = fma(z, Float64(1.6453555072203998 * Float64(y * a)), fma(y, Float64(b * 1.6453555072203998), x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(3.13060547623 - N[(t / N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -4.2e-8], t$95$1, If[LessEqual[z, 4.6e-5], N[(z * N[(1.6453555072203998 * N[(y * a), $MachinePrecision]), $MachinePrecision] + N[(y * N[(b * 1.6453555072203998), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.19999999999999989e-8 or 4.6e-5 < z

   1. Initial program 28.3%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 83.1%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{y \cdot \left(\frac{313060547623}{100000000000} - \left(-1 \cdot \frac{\frac{45796100221584283915100827016327}{100000000000000000000000000000} + t}{{z}^{2}} + \frac{3652704169880641883561}{100000000000000000000} \cdot \frac{1}{z}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 92.4%

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{3.13060547623 - \left(\frac{36.52704169880642}{z} - \frac{t + 457.9610022158428}{z \cdot z}\right)}, x\right) \]

   9. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000} - -1 \cdot \frac{t}{\color{blue}{{z}^{2}}}, x\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 92.4%

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

   if -4.19999999999999989e-8 < z < 4.6e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 89.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(z, 1.6453555072203998 \cdot \left(y \cdot a\right), \mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623 - \frac{t}{z \cdot \left(-z\right)}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;x + 1.6453555072203998 \cdot \left(\mathsf{fma}\left(z, t, a\right) \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 60000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -4.1e+37)
   (fma y 3.13060547623 x)
   (if (<= z -1.2e-71)
     (+ x (* 1.6453555072203998 (* (fma z t a) (* y z))))
     (if (<= z 60000000.0)
       (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 <= -4.1e+37) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= -1.2e-71) {
		tmp = x + (1.6453555072203998 * (fma(z, t, a) * (y * z)));
	} else if (z <= 60000000.0) {
		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 <= -4.1e+37)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= -1.2e-71)
		tmp = Float64(x + Float64(1.6453555072203998 * Float64(fma(z, t, a) * Float64(y * z))));
	elseif (z <= 60000000.0)
		tmp = fma(Float64(y * b), 1.6453555072203998, x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -4.1e+37], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, -1.2e-71], N[(x + N[(1.6453555072203998 * N[(N[(z * t + a), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 60000000.0], N[(N[(y * b), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.0999999999999998e37 or 6e7 < z

   1. Initial program 19.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 93.2

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

   5. Applied rewrites 93.2%

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

   if -4.0999999999999998e37 < z < -1.2e-71

   1. Initial program 93.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right) \cdot \frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}}} \]

      3. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\frac{1}{\left(\left(\left(z + \frac{15234687407}{1000000000}\right) \cdot z + \frac{314690115749}{10000000000}\right) \cdot z + \frac{119400905721}{10000000000}\right) \cdot z + \frac{607771387771}{1000000000000}} \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(y, a, y \cdot \left(z \cdot t\right)\right), y \cdot b\right)} \]

   7. Applied rewrites 90.0%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 75.7%

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

   11. Taylor expanded in b around 0

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

   13. Applied rewrites 65.6%

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

   if -1.2e-71 < z < 6e7

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 43.7

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 86.3

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

   8. Applied rewrites 86.3%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+37}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;x + 1.6453555072203998 \cdot \left(\mathsf{fma}\left(z, t, a\right) \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 60000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(1.6453555072203998, a \cdot \left(y \cdot z\right), x\right)\\ \mathbf{elif}\;z \leq 60000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -6.5e+50)
   (fma y 3.13060547623 x)
   (if (<= z -1.2e-71)
     (fma 1.6453555072203998 (* a (* y z)) x)
     (if (<= z 60000000.0)
       (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 <= -6.5e+50) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= -1.2e-71) {
		tmp = fma(1.6453555072203998, (a * (y * z)), x);
	} else if (z <= 60000000.0) {
		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 <= -6.5e+50)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= -1.2e-71)
		tmp = fma(1.6453555072203998, Float64(a * Float64(y * z)), x);
	elseif (z <= 60000000.0)
		tmp = fma(Float64(y * b), 1.6453555072203998, x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -6.5e+50], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, -1.2e-71], N[(1.6453555072203998 * N[(a * N[(y * z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 60000000.0], N[(N[(y * b), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5000000000000003e50 or 6e7 < z

   1. Initial program 17.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 94.0

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

   5. Applied rewrites 94.0%

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

   if -6.5000000000000003e50 < z < -1.2e-71

   1. Initial program 94.3%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-*r* N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. associate-*r* N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 54.1%

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

   if -1.2e-71 < z < 6e7

   1. Initial program 99.6%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 43.7

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

   5. Applied rewrites 43.7%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 86.3

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

   8. Applied rewrites 86.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;\mathsf{fma}\left(1.6453555072203998, a \cdot \left(y \cdot z\right), x\right)\\ \mathbf{elif}\;z \leq 60000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 83.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 60000000:\\ \;\;\;\;\mathsf{fma}\left(y \cdot b, 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.15e+21)
   (fma y 3.13060547623 x)
   (if (<= z 60000000.0)
     (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 <= -2.15e+21) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 60000000.0) {
		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 <= -2.15e+21)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 60000000.0)
		tmp = fma(Float64(y * b), 1.6453555072203998, x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.15e+21], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 60000000.0], N[(N[(y * b), $MachinePrecision] * 1.6453555072203998 + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.15e21 or 6e7 < z

   1. Initial program 23.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 89.6

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

   5. Applied rewrites 89.6%

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

   if -2.15e21 < z < 6e7

   1. Initial program 98.9%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 38.4

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

   5. Applied rewrites 38.4%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 76.9

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

   8. Applied rewrites 76.9%

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

Alternative 16: 83.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 60000000:\\ \;\;\;\;\mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.15e+21)
   (fma y 3.13060547623 x)
   (if (<= z 60000000.0)
     (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 <= -2.15e+21) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 60000000.0) {
		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 <= -2.15e+21)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 60000000.0)
		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, -2.15e+21], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 60000000.0], 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 < -2.15e21 or 6e7 < z

   1. Initial program 23.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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 89.6

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

   5. Applied rewrites 89.6%

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

   if -2.15e21 < z < 6e7

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 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 76.9

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

   5. Applied rewrites 76.9%

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

Alternative 17: 62.2% 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 62.8%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 62.8

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

5. Applied rewrites 62.8%

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

Alternative 18: 21.8% 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 62.8%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 62.8

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

5. Applied rewrites 62.8%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 20.1%

   \[\leadsto y \cdot \color{blue}{3.13060547623} \]
9. Add Preprocessing

Developer Target 1: 98.6% 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 2024228 
(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))))