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

Percentage Accurate: 59.2% → 97.1%

Time: 16.8s

Alternatives: 12

Speedup: 3.3×

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: 59.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-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 egg-rr 98.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 98.6%

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

4. Final simplification 98.4%

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

Alternative 2: 71.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(y \cdot 1.6453555072203998\right)\\ t_2 := \frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+151}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* b (* y 1.6453555072203998)))
        (t_2
         (/
          (*
           y
           (+
            (* z (+ (* z (+ (* z (+ (* z 3.13060547623) 11.1667541262)) t)) a))
            b))
          (+
           (*
            z
            (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
           0.607771387771))))
   (if (<= t_2 -2e+149)
     t_1
     (if (<= t_2 1e+151)
       x
       (if (<= t_2 INFINITY) t_1 (fma y 3.13060547623 x))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = b * (y * 1.6453555072203998);
	double t_2 = (y * ((z * ((z * ((z * ((z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / ((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771);
	double tmp;
	if (t_2 <= -2e+149) {
		tmp = t_1;
	} else if (t_2 <= 1e+151) {
		tmp = x;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(b * Float64(y * 1.6453555072203998))
	t_2 = Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771))
	tmp = 0.0
	if (t_2 <= -2e+149)
		tmp = t_1;
	elseif (t_2 <= 1e+151)
		tmp = x;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(b * N[(y * 1.6453555072203998), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * N[(N[(z * N[(N[(z * N[(N[(z * N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(N[(z * N[(N[(z * N[(z + 15.234687407), $MachinePrecision]), $MachinePrecision] + 31.4690115749), $MachinePrecision]), $MachinePrecision] + 11.9400905721), $MachinePrecision]), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+149], t$95$1, If[LessEqual[t$95$2, 1e+151], x, If[LessEqual[t$95$2, Infinity], t$95$1, N[(y * 3.13060547623 + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < -2.0000000000000001e149 or 1.00000000000000002e151 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 313060547623/100000000000 binary64)) #s(literal 55833770631/5000000000 binary64)) z) t) z) a) z) b)) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 z #s(literal 15234687407/1000000000 binary64)) z) #s(literal 314690115749/10000000000 binary64)) z) #s(literal 119400905721/10000000000 binary64)) z) #s(literal 607771387771/1000000000000 binary64))) < +inf.0

   1. Initial program 90.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.3%

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

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{313060547623}{100000000000}, \frac{55833770631}{5000000000}\right), t\right), a\right), b\right), \color{blue}{\frac{1000000000000}{607771387771} \cdot \frac{y}{x}}, 1\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{313060547623}{100000000000}, \frac{55833770631}{5000000000}\right), t\right), a\right), b\right), \color{blue}{\frac{\frac{1000000000000}{607771387771} \cdot y}{x}}, 1\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{313060547623}{100000000000}, \frac{55833770631}{5000000000}\right), t\right), a\right), b\right), \color{blue}{\frac{\frac{1000000000000}{607771387771} \cdot y}{x}}, 1\right) \]

      3. *-lowering-*.f64 68.1

         \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), \frac{\color{blue}{1.6453555072203998 \cdot y}}{x}, 1\right) \]

   8. Simplified 68.1%

      \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), \color{blue}{\frac{1.6453555072203998 \cdot y}{x}}, 1\right) \]

   9. Taylor expanded in b around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-commutative N/A

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

       6. *-lowering-*.f64 61.8

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

   11. Simplified 61.8%

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

   if -2.0000000000000001e149 < (/.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))) < 1.00000000000000002e151

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 65.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 96.2

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

   5. Simplified 96.2%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq -2 \cdot 10^{+149}:\\ \;\;\;\;b \cdot \left(y \cdot 1.6453555072203998\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq 10^{+151}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq \infty:\\ \;\;\;\;b \cdot \left(y \cdot 1.6453555072203998\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 96.4%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-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 egg-rr 98.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in z around -inf

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

   5. Simplified 90.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right) - \frac{\frac{\mathsf{fma}\left(y, t, \mathsf{fma}\left(y, -98.5170599679272, y \cdot 556.47806218377\right)\right)}{0 - z} + y \cdot 36.52704169880642}{z}} \]

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot y}{{z}^{2}}\right)\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \color{blue}{\left(0 - \frac{t \cdot y}{{z}^{2}}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \color{blue}{\left(0 - \frac{t \cdot y}{{z}^{2}}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - \color{blue}{t \cdot \frac{y}{{z}^{2}}}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - \color{blue}{t \cdot \frac{y}{{z}^{2}}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - t \cdot \color{blue}{\frac{y}{{z}^{2}}}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - t \cdot \frac{y}{\color{blue}{z \cdot z}}\right) \]

      8. *-lowering-*.f64 98.6

         \[\leadsto \mathsf{fma}\left(y, 3.13060547623, x\right) - \left(0 - t \cdot \frac{y}{\color{blue}{z \cdot z}}\right) \]

   8. Simplified 98.6%

      \[\leadsto \mathsf{fma}\left(y, 3.13060547623, x\right) - \color{blue}{\left(0 - t \cdot \frac{y}{z \cdot z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

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

Alternative 4: 62.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq 2 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

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))
      2e-62)
   x
   (fma y 3.13060547623 x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (((y * ((z * ((z * ((z * ((z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / ((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771)) <= 2e-62) {
		tmp = x;
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * 3.13060547623) + 11.1667541262)) + t)) + a)) + b)) / Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771)) <= 2e-62)
		tmp = x;
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[N[(N[(y * N[(N[(z * N[(N[(z * N[(N[(z * N[(N[(z * 3.13060547623), $MachinePrecision] + 11.1667541262), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(N[(z * N[(N[(z * N[(z + 15.234687407), $MachinePrecision]), $MachinePrecision] + 31.4690115749), $MachinePrecision]), $MachinePrecision] + 11.9400905721), $MachinePrecision]), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision], 2e-62], x, 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))) < 2.0000000000000001e-62

   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. Taylor expanded in x around inf

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

   5. Simplified 55.6%

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

   if 2.0000000000000001e-62 < (/.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 32.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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. accelerator-lowering-fma.f64 74.2

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

   5. Simplified 74.2%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z \cdot 3.13060547623 + 11.1667541262\right) + t\right) + a\right) + b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771} \leq 2 \cdot 10^{-62}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623, x\right) + t \cdot \frac{y}{z \cdot z}\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+34}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, t, a\right), b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (fma y 3.13060547623 x) (* t (/ y (* z z))))))
   (if (<= z -1.35e+32)
     t_1
     (if (<= z 4.2e+34)
       (+
        x
        (/
         (* y (fma z (fma z t a) b))
         (+
          (*
           z
           (+ (* z (+ (* z (+ z 15.234687407)) 31.4690115749)) 11.9400905721))
          0.607771387771)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, 3.13060547623, x) + (t * (y / (z * z)));
	double tmp;
	if (z <= -1.35e+32) {
		tmp = t_1;
	} else if (z <= 4.2e+34) {
		tmp = x + ((y * fma(z, fma(z, t, a), b)) / ((z * ((z * ((z * (z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(y, 3.13060547623, x) + Float64(t * Float64(y / Float64(z * z))))
	tmp = 0.0
	if (z <= -1.35e+32)
		tmp = t_1;
	elseif (z <= 4.2e+34)
		tmp = Float64(x + Float64(Float64(y * fma(z, fma(z, t, a), b)) / Float64(Float64(z * Float64(Float64(z * Float64(Float64(z * Float64(z + 15.234687407)) + 31.4690115749)) + 11.9400905721)) + 0.607771387771)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * 3.13060547623 + x), $MachinePrecision] + N[(t * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.35e+32], t$95$1, If[LessEqual[z, 4.2e+34], N[(x + N[(N[(y * N[(z * N[(z * t + a), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(N[(z * N[(N[(z * N[(z + 15.234687407), $MachinePrecision]), $MachinePrecision] + 31.4690115749), $MachinePrecision]), $MachinePrecision] + 11.9400905721), $MachinePrecision]), $MachinePrecision] + 0.607771387771), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35000000000000006e32 or 4.20000000000000035e34 < z

   1. Initial program 8.2%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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. Simplified 91.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right) - \frac{\frac{\mathsf{fma}\left(y, t, \mathsf{fma}\left(y, -98.5170599679272, y \cdot 556.47806218377\right)\right)}{0 - z} + y \cdot 36.52704169880642}{z}} \]

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot y}{{z}^{2}}\right)\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \color{blue}{\left(0 - \frac{t \cdot y}{{z}^{2}}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \color{blue}{\left(0 - \frac{t \cdot y}{{z}^{2}}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - \color{blue}{t \cdot \frac{y}{{z}^{2}}}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - \color{blue}{t \cdot \frac{y}{{z}^{2}}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - t \cdot \color{blue}{\frac{y}{{z}^{2}}}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - t \cdot \frac{y}{\color{blue}{z \cdot z}}\right) \]

      8. *-lowering-*.f64 98.1

         \[\leadsto \mathsf{fma}\left(y, 3.13060547623, x\right) - \left(0 - t \cdot \frac{y}{\color{blue}{z \cdot z}}\right) \]

   8. Simplified 98.1%

      \[\leadsto \mathsf{fma}\left(y, 3.13060547623, x\right) - \color{blue}{\left(0 - t \cdot \frac{y}{z \cdot z}\right)} \]

   if -1.35000000000000006e32 < z < 4.20000000000000035e34

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 98.3

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

   5. Simplified 98.3%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right) + t \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+34}:\\ \;\;\;\;x + \frac{y \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, t, a\right), b\right)}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.234687407\right) + 31.4690115749\right) + 11.9400905721\right) + 0.607771387771}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right) + t \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623, x\right) + t \cdot \frac{y}{z \cdot z}\\ \mathbf{if}\;z \leq -2000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 1.6453555072203998, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), t\right), a\right), b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (fma y 3.13060547623 x) (* t (/ y (* z z))))))
   (if (<= z -2000000000.0)
     t_1
     (if (<= z 2.45e+32)
       (fma
        (* y 1.6453555072203998)
        (fma z (fma z (fma z (fma 3.13060547623 z 11.1667541262) t) a) b)
        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, x) + (t * (y / (z * z)));
	double tmp;
	if (z <= -2000000000.0) {
		tmp = t_1;
	} else if (z <= 2.45e+32) {
		tmp = fma((y * 1.6453555072203998), fma(z, fma(z, fma(z, fma(3.13060547623, z, 11.1667541262), t), a), b), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(y, 3.13060547623, x) + Float64(t * Float64(y / Float64(z * z))))
	tmp = 0.0
	if (z <= -2000000000.0)
		tmp = t_1;
	elseif (z <= 2.45e+32)
		tmp = fma(Float64(y * 1.6453555072203998), fma(z, fma(z, fma(z, fma(3.13060547623, z, 11.1667541262), t), a), b), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y * 3.13060547623 + x), $MachinePrecision] + N[(t * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2000000000.0], t$95$1, If[LessEqual[z, 2.45e+32], N[(N[(y * 1.6453555072203998), $MachinePrecision] * N[(z * N[(z * N[(z * N[(3.13060547623 * z + 11.1667541262), $MachinePrecision] + t), $MachinePrecision] + a), $MachinePrecision] + b), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e9 or 2.4500000000000001e32 < z

   1. Initial program 11.0%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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. Simplified 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right) - \frac{\frac{\mathsf{fma}\left(y, t, \mathsf{fma}\left(y, -98.5170599679272, y \cdot 556.47806218377\right)\right)}{0 - z} + y \cdot 36.52704169880642}{z}} \]

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot y}{{z}^{2}}\right)\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \color{blue}{\left(0 - \frac{t \cdot y}{{z}^{2}}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \color{blue}{\left(0 - \frac{t \cdot y}{{z}^{2}}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - \color{blue}{t \cdot \frac{y}{{z}^{2}}}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - \color{blue}{t \cdot \frac{y}{{z}^{2}}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - t \cdot \color{blue}{\frac{y}{{z}^{2}}}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - t \cdot \frac{y}{\color{blue}{z \cdot z}}\right) \]

      8. *-lowering-*.f64 96.5

         \[\leadsto \mathsf{fma}\left(y, 3.13060547623, x\right) - \left(0 - t \cdot \frac{y}{\color{blue}{z \cdot z}}\right) \]

   8. Simplified 96.5%

      \[\leadsto \mathsf{fma}\left(y, 3.13060547623, x\right) - \color{blue}{\left(0 - t \cdot \frac{y}{z \cdot z}\right)} \]

   if -2e9 < z < 2.4500000000000001e32

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 88.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{313060547623}{100000000000}, \frac{55833770631}{5000000000}\right), t\right), a\right), b\right), \color{blue}{\frac{1000000000000}{607771387771} \cdot \frac{y}{x}}, 1\right) \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{313060547623}{100000000000}, \frac{55833770631}{5000000000}\right), t\right), a\right), b\right), \color{blue}{\frac{\frac{1000000000000}{607771387771} \cdot y}{x}}, 1\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{313060547623}{100000000000}, \frac{55833770631}{5000000000}\right), t\right), a\right), b\right), \color{blue}{\frac{\frac{1000000000000}{607771387771} \cdot y}{x}}, 1\right) \]

      3. *-lowering-*.f64 85.8

         \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), \frac{\color{blue}{1.6453555072203998 \cdot y}}{x}, 1\right) \]

   8. Simplified 85.8%

      \[\leadsto x \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 3.13060547623, 11.1667541262\right), t\right), a\right), b\right), \color{blue}{\frac{1.6453555072203998 \cdot y}{x}}, 1\right) \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. accelerator-lowering-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. *-lowering-*.f64 N/A

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

       6. +-commutative N/A

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

       7. accelerator-lowering-fma.f64 N/A

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

       8. +-commutative N/A

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

       9. accelerator-lowering-fma.f64 N/A

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

       10. +-commutative N/A

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

       11. accelerator-lowering-fma.f64 N/A

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

       12. +-commutative N/A

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

       13. accelerator-lowering-fma.f64 97.5

           \[\leadsto \mathsf{fma}\left(y \cdot 1.6453555072203998, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(3.13060547623, z, 11.1667541262\right)}, t\right), a\right), b\right), x\right) \]

   11. Simplified 97.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot 1.6453555072203998, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), t\right), a\right), b\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2000000000:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right) + t \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot 1.6453555072203998, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(3.13060547623, z, 11.1667541262\right), t\right), a\right), b\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right) + t \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 3.13060547623, x\right) + t \cdot \frac{y}{z \cdot z}\\ \mathbf{if}\;z \leq -0.0028:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \left(a \cdot 1.6453555072203998\right), \mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (fma y 3.13060547623 x) (* t (/ y (* z z))))))
   (if (<= z -0.0028)
     t_1
     (if (<= z 2.45e+32)
       (fma
        z
        (* y (* a 1.6453555072203998))
        (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, x) + (t * (y / (z * z)));
	double tmp;
	if (z <= -0.0028) {
		tmp = t_1;
	} else if (z <= 2.45e+32) {
		tmp = fma(z, (y * (a * 1.6453555072203998)), fma(y, (b * 1.6453555072203998), x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(fma(y, 3.13060547623, x) + Float64(t * Float64(y / Float64(z * z))))
	tmp = 0.0
	if (z <= -0.0028)
		tmp = t_1;
	elseif (z <= 2.45e+32)
		tmp = fma(z, Float64(y * Float64(a * 1.6453555072203998)), 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[(N[(y * 3.13060547623 + x), $MachinePrecision] + N[(t * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.0028], t$95$1, If[LessEqual[z, 2.45e+32], N[(z * N[(y * N[(a * 1.6453555072203998), $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 < -0.00279999999999999997 or 2.4500000000000001e32 < z

   1. Initial program 14.3%

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

   3. Taylor expanded in z around -inf

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

   5. Simplified 88.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 3.13060547623, x\right) - \frac{\frac{\mathsf{fma}\left(y, t, \mathsf{fma}\left(y, -98.5170599679272, y \cdot 556.47806218377\right)\right)}{0 - z} + y \cdot 36.52704169880642}{z}} \]

   6. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot y}{{z}^{2}}\right)\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \color{blue}{\left(0 - \frac{t \cdot y}{{z}^{2}}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \color{blue}{\left(0 - \frac{t \cdot y}{{z}^{2}}\right)} \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - \color{blue}{t \cdot \frac{y}{{z}^{2}}}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - \color{blue}{t \cdot \frac{y}{{z}^{2}}}\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - t \cdot \color{blue}{\frac{y}{{z}^{2}}}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(y, \frac{313060547623}{100000000000}, x\right) - \left(0 - t \cdot \frac{y}{\color{blue}{z \cdot z}}\right) \]

      8. *-lowering-*.f64 95.2

         \[\leadsto \mathsf{fma}\left(y, 3.13060547623, x\right) - \left(0 - t \cdot \frac{y}{\color{blue}{z \cdot z}}\right) \]

   8. Simplified 95.2%

      \[\leadsto \mathsf{fma}\left(y, 3.13060547623, x\right) - \color{blue}{\left(0 - t \cdot \frac{y}{z \cdot z}\right)} \]

   if -0.00279999999999999997 < z < 2.4500000000000001e32

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right) + z \cdot \left(\frac{1000000000000}{607771387771} \cdot \left(a \cdot y\right) - \frac{11940090572100000000000000}{369386059793087248348441} \cdot \left(b \cdot y\right)\right)\right)} \]
   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 82.0

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

   5. Simplified 82.0%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 86.1

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

   8. Simplified 86.1%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0028:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right) + t \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \left(a \cdot 1.6453555072203998\right), \mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right) + t \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(z, y \cdot \left(a \cdot 1.6453555072203998\right), \mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.5e+28)
   (fma y 3.13060547623 x)
   (if (<= z 1.9e+34)
     (fma z (* y (* a 1.6453555072203998)) (fma y (* b 1.6453555072203998) x))
     (fma y 3.13060547623 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.5e+28) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.9e+34) {
		tmp = fma(z, (y * (a * 1.6453555072203998)), fma(y, (b * 1.6453555072203998), x));
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.5e+28)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.9e+34)
		tmp = fma(z, Float64(y * Float64(a * 1.6453555072203998)), fma(y, Float64(b * 1.6453555072203998), x));
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.5e+28], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.9e+34], N[(z * N[(y * N[(a * 1.6453555072203998), $MachinePrecision]), $MachinePrecision] + N[(y * N[(b * 1.6453555072203998), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.5e28 or 1.9000000000000001e34 < z

   1. Initial program 8.9%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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. accelerator-lowering-fma.f64 95.2

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

   5. Simplified 95.2%

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

   if -1.5e28 < z < 1.9000000000000001e34

   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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.f64 78.7

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

   5. Simplified 78.7%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 82.6

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

   8. Simplified 82.6%

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

Alternative 9: 83.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-16}:\\ \;\;\;\;x + 1.6453555072203998 \cdot \left(y \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.7e+28)
   (fma y 3.13060547623 x)
   (if (<= z 1.3e-16)
     (+ x (* 1.6453555072203998 (* y b)))
     (fma y 3.13060547623 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.7e+28) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.3e-16) {
		tmp = x + (1.6453555072203998 * (y * b));
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.7e+28)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.3e-16)
		tmp = Float64(x + Float64(1.6453555072203998 * Float64(y * b)));
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.7e+28], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.3e-16], N[(x + N[(1.6453555072203998 * N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7000000000000002e28 or 1.2999999999999999e-16 < z

   1. Initial program 12.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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. accelerator-lowering-fma.f64 93.9

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

   5. Simplified 93.9%

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

   if -2.7000000000000002e28 < z < 1.2999999999999999e-16

   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

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

      1. *-lowering-*.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 78.2

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

   5. Simplified 78.2%

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

Alternative 10: 83.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-16}:\\ \;\;\;\;\mathsf{fma}\left(y, b \cdot 1.6453555072203998, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 3.13060547623, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.55e+28)
   (fma y 3.13060547623 x)
   (if (<= z 1.3e-16)
     (fma y (* b 1.6453555072203998) x)
     (fma y 3.13060547623 x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.55e+28) {
		tmp = fma(y, 3.13060547623, x);
	} else if (z <= 1.3e-16) {
		tmp = fma(y, (b * 1.6453555072203998), x);
	} else {
		tmp = fma(y, 3.13060547623, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.55e+28)
		tmp = fma(y, 3.13060547623, x);
	elseif (z <= 1.3e-16)
		tmp = fma(y, Float64(b * 1.6453555072203998), x);
	else
		tmp = fma(y, 3.13060547623, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.55e+28], N[(y * 3.13060547623 + x), $MachinePrecision], If[LessEqual[z, 1.3e-16], N[(y * N[(b * 1.6453555072203998), $MachinePrecision] + x), $MachinePrecision], N[(y * 3.13060547623 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.55e28 or 1.2999999999999999e-16 < z

   1. Initial program 12.4%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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. accelerator-lowering-fma.f64 93.9

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

   5. Simplified 93.9%

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

   if -1.55e28 < z < 1.2999999999999999e-16

   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

      \[\leadsto \color{blue}{x + \frac{1000000000000}{607771387771} \cdot \left(b \cdot y\right)} \]
   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. accelerator-lowering-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. *-lowering-*.f64 78.2

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

   5. Simplified 78.2%

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

Alternative 11: 49.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{+224}:\\ \;\;\;\;y \cdot 3.13060547623\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 3.13060547623\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.05e+224)
   (* y 3.13060547623)
   (if (<= y 2.2e+60) x (* y 3.13060547623))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.05e+224) {
		tmp = y * 3.13060547623;
	} else if (y <= 2.2e+60) {
		tmp = x;
	} else {
		tmp = y * 3.13060547623;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-3.05d+224)) then
        tmp = y * 3.13060547623d0
    else if (y <= 2.2d+60) then
        tmp = x
    else
        tmp = y * 3.13060547623d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.05e+224) {
		tmp = y * 3.13060547623;
	} else if (y <= 2.2e+60) {
		tmp = x;
	} else {
		tmp = y * 3.13060547623;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -3.05e+224:
		tmp = y * 3.13060547623
	elif y <= 2.2e+60:
		tmp = x
	else:
		tmp = y * 3.13060547623
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.05e+224)
		tmp = Float64(y * 3.13060547623);
	elseif (y <= 2.2e+60)
		tmp = x;
	else
		tmp = Float64(y * 3.13060547623);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -3.05e+224)
		tmp = y * 3.13060547623;
	elseif (y <= 2.2e+60)
		tmp = x;
	else
		tmp = y * 3.13060547623;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.05e+224], N[(y * 3.13060547623), $MachinePrecision], If[LessEqual[y, 2.2e+60], x, N[(y * 3.13060547623), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.05e224 or 2.19999999999999996e60 < y

   1. Initial program 52.6%

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.13060547623 + 11.1667541262\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.234687407\right) \cdot z + 31.4690115749\right) \cdot z + 11.9400905721\right) \cdot z + 0.607771387771} \]
   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. accelerator-lowering-fma.f64 56.3

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

   5. Simplified 56.3%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 44.6

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

   8. Simplified 44.6%

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

   if -3.05e224 < y < 2.19999999999999996e60

   1. Initial program 55.7%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 61.1%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{+224}:\\ \;\;\;\;y \cdot 3.13060547623\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+60}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot 3.13060547623\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.5% accurate, 79.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.0%

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

3. Taylor expanded in x around inf

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

5. Simplified 50.3%

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

Developer Target 1: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024196 
(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))))