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

Percentage Accurate: 69.4% → 99.2%

Time: 10.1s

Alternatives: 11

Speedup: 2.5×

Specification

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+
     (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
     0.279195317918525))
   (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))

C

double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y * ((((z * 0.0692910599291889d0) + 0.4917317610505968d0) * z) + 0.279195317918525d0)) / (((z + 6.012459259764103d0) * z) + 3.350343815022304d0))
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

Python

def code(x, y, z):
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+
  x
  (/
   (*
    y
    (+
     (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
     0.279195317918525))
   (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))

C

double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y * ((((z * 0.0692910599291889d0) + 0.4917317610505968d0) * z) + 0.279195317918525d0)) / (((z + 6.012459259764103d0) * z) + 3.350343815022304d0))
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
}

Python

def code(x, y, z):
	return x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{elif}\;z \leq 4.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0007936505811533442, z, -0.00277777777751721\right) \cdot y, z, \mathsf{fma}\left(y, 0.08333333333333323, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)
   (if (<= z 4.4)
     (fma
      (* (fma 0.0007936505811533442 z -0.00277777777751721) y)
      z
      (fma y 0.08333333333333323 x))
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	} else if (z <= 4.4) {
		tmp = fma((fma(0.0007936505811533442, z, -0.00277777777751721) * y), z, fma(y, 0.08333333333333323, x));
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x);
	elseif (z <= 4.4)
		tmp = fma(Float64(fma(0.0007936505811533442, z, -0.00277777777751721) * y), z, fma(y, 0.08333333333333323, x));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 4.4], N[(N[(N[(0.0007936505811533442 * z + -0.00277777777751721), $MachinePrecision] * y), $MachinePrecision] * z + N[(y * 0.08333333333333323 + x), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 28.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right)\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right)} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \left(\frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)} \]

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

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \color{blue}{\frac{y}{z} \cdot \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)} \]

      4. metadata-eval N/A

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{y}{z} \cdot \color{blue}{\frac{751220861604756070699018739433}{10000000000000000000000000000000}} \]

      5. metadata-eval N/A

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{y}{z} \cdot \color{blue}{\frac{\frac{-751220861604756070699018739433}{10000000000000000000000000000000}}{-1}} \]

      6. metadata-eval N/A

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{y}{z} \cdot \frac{\color{blue}{\frac{-307332350656623}{625000000000000} - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}}}{-1} \]

      7. times-frac N/A

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \color{blue}{\frac{y \cdot \left(\frac{-307332350656623}{625000000000000} - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right)}{z \cdot -1}} \]

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

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{\color{blue}{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}}{z \cdot -1} \]

      9. *-commutative N/A

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{\color{blue}{-1 \cdot z}} \]

      10. mul-1-neg N/A

          \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}\right)\right)} \]

      12. mul-1-neg N/A

          \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \color{blue}{-1 \cdot \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}} \]

      13. associate-+r+ N/A

          \[\leadsto \color{blue}{x + \left(\frac{692910599291889}{10000000000000000} \cdot y + -1 \cdot \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}\right)} \]

      14. +-commutative N/A

          \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} + \frac{692910599291889}{10000000000000000} \cdot y\right)} \]

      15. +-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} + \frac{692910599291889}{10000000000000000} \cdot y\right) + x} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)} \]

   if -5.4000000000000004 < z < 4.4000000000000004

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\left(\frac{307332350656623}{2093964884388940} \cdot y + z \cdot \left(\frac{692910599291889}{33503438150223040} \cdot y - \left(\frac{272651677654809570312500000}{10961722342634967150292985809} \cdot y + \frac{6012459259764103}{3350343815022304} \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)\right)\right) - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + z \cdot \left(\left(\frac{307332350656623}{2093964884388940} \cdot y + z \cdot \left(\frac{692910599291889}{33503438150223040} \cdot y - \left(\frac{272651677654809570312500000}{10961722342634967150292985809} \cdot y + \frac{6012459259764103}{3350343815022304} \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)\right)\right) - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\left(\frac{307332350656623}{2093964884388940} \cdot y + z \cdot \left(\frac{692910599291889}{33503438150223040} \cdot y - \left(\frac{272651677654809570312500000}{10961722342634967150292985809} \cdot y + \frac{6012459259764103}{3350343815022304} \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)\right)\right) - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right) + \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right)} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{307332350656623}{2093964884388940} \cdot y + z \cdot \left(\frac{692910599291889}{33503438150223040} \cdot y - \left(\frac{272651677654809570312500000}{10961722342634967150292985809} \cdot y + \frac{6012459259764103}{3350343815022304} \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)\right)\right) - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right) \cdot z} + \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{307332350656623}{2093964884388940} \cdot y + z \cdot \left(\frac{692910599291889}{33503438150223040} \cdot y - \left(\frac{272651677654809570312500000}{10961722342634967150292985809} \cdot y + \frac{6012459259764103}{3350343815022304} \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)\right)\right) - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y, z, x + \frac{279195317918525}{3350343815022304} \cdot y\right)} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.00277777777751721, y, \left(y \cdot 0.0007936505811533442\right) \cdot z\right), z, \mathsf{fma}\left(y, 0.08333333333333323, x\right)\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

      \[\leadsto \mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.0007936505811533442, z, -0.00277777777751721\right), z, \mathsf{fma}\left(y, 0.08333333333333323, x\right)\right) \]

   if 4.4000000000000004 < z

   1. Initial program 49.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{elif}\;z \leq 4.4:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0007936505811533442, z, -0.00277777777751721\right) \cdot y, z, \mathsf{fma}\left(y, 0.08333333333333323, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(0.279195317918525 + \left(0.4917317610505968 + 0.0692910599291889 \cdot z\right) \cdot z\right) \cdot y}{3.350343815022304 + \left(6.012459259764103 + z\right) \cdot z}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{+96}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           (+
            0.279195317918525
            (* (+ 0.4917317610505968 (* 0.0692910599291889 z)) z))
           y)
          (+ 3.350343815022304 (* (+ 6.012459259764103 z) z)))))
   (if (<= t_0 (- INFINITY))
     (fma 0.0692910599291889 y x)
     (if (<= t_0 -4e+96)
       (* 0.08333333333333323 y)
       (fma 0.0692910599291889 y x)))))

C

double code(double x, double y, double z) {
	double t_0 = ((0.279195317918525 + ((0.4917317610505968 + (0.0692910599291889 * z)) * z)) * y) / (3.350343815022304 + ((6.012459259764103 + z) * z));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (t_0 <= -4e+96) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(0.279195317918525 + Float64(Float64(0.4917317610505968 + Float64(0.0692910599291889 * z)) * z)) * y) / Float64(3.350343815022304 + Float64(Float64(6.012459259764103 + z) * z)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(0.0692910599291889, y, x);
	elseif (t_0 <= -4e+96)
		tmp = Float64(0.08333333333333323 * y);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(0.279195317918525 + N[(N[(0.4917317610505968 + N[(0.0692910599291889 * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(3.350343815022304 + N[(N[(6.012459259764103 + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[t$95$0, -4e+96], N[(0.08333333333333323 * y), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -inf.0 or -4.0000000000000002e96 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))

   1. Initial program 68.4%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 81.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 81.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   if -inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -4.0000000000000002e96

   1. Initial program 99.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{279195317918525}{3350343815022304} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{279195317918525}{3350343815022304}} + x \]

      3. lower-fma.f64 94.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.08333333333333323, x\right)} \]

   5. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.08333333333333323, x\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{279195317918525}{3350343815022304} \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.7%

      \[\leadsto y \cdot \color{blue}{0.08333333333333323} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(0.279195317918525 + \left(0.4917317610505968 + 0.0692910599291889 \cdot z\right) \cdot z\right) \cdot y}{3.350343815022304 + \left(6.012459259764103 + z\right) \cdot z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;\frac{\left(0.279195317918525 + \left(0.4917317610505968 + 0.0692910599291889 \cdot z\right) \cdot z\right) \cdot y}{3.350343815022304 + \left(6.012459259764103 + z\right) \cdot z} \leq -4 \cdot 10^{+96}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(0.279195317918525 + \left(0.4917317610505968 + 0.0692910599291889 \cdot z\right) \cdot z\right) \cdot y}{3.350343815022304 + \left(6.012459259764103 + z\right) \cdot z} \leq 10^{+302}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\frac{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (+
         0.279195317918525
         (* (+ 0.4917317610505968 (* 0.0692910599291889 z)) z))
        y)
       (+ 3.350343815022304 (* (+ 6.012459259764103 z) z)))
      1e+302)
   (+
    (/
     (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
     (/ (fma (+ 6.012459259764103 z) z 3.350343815022304) y))
    x)
   (+ (* 0.0692910599291889 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((0.279195317918525 + ((0.4917317610505968 + (0.0692910599291889 * z)) * z)) * y) / (3.350343815022304 + ((6.012459259764103 + z) * z))) <= 1e+302) {
		tmp = (fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / (fma((6.012459259764103 + z), z, 3.350343815022304) / y)) + x;
	} else {
		tmp = (0.0692910599291889 * y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(0.279195317918525 + Float64(Float64(0.4917317610505968 + Float64(0.0692910599291889 * z)) * z)) * y) / Float64(3.350343815022304 + Float64(Float64(6.012459259764103 + z) * z))) <= 1e+302)
		tmp = Float64(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / Float64(fma(Float64(6.012459259764103 + z), z, 3.350343815022304) / y)) + x);
	else
		tmp = Float64(Float64(0.0692910599291889 * y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(0.279195317918525 + N[(N[(0.4917317610505968 + N[(0.0692910599291889 * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(3.350343815022304 + N[(N[(6.012459259764103 + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+302], N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] / N[(N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(0.0692910599291889 * y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 1.0000000000000001e302

   1. Initial program 95.3%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      3. lower-*.f64 95.3

         \[\leadsto x + \frac{\color{blue}{\left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right) \cdot y}}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      4. lift-+.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)} \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      5. lift-*.f64 N/A

         \[\leadsto x + \frac{\left(\color{blue}{\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z} + \frac{11167812716741}{40000000000000}\right) \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      6. lower-fma.f64 95.3

         \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(z \cdot 0.0692910599291889 + 0.4917317610505968, z, 0.279195317918525\right)} \cdot y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      7. lift-+.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      8. lift-*.f64 N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{z \cdot \frac{692910599291889}{10000000000000000}} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      9. *-commutative N/A

         \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} \cdot z} + \frac{307332350656623}{625000000000000}, z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      10. lower-fma.f64 95.3

          \[\leadsto x + \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right)}, z, 0.279195317918525\right) \cdot y}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]

      11. lift-+.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      12. lift-*.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\color{blue}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z} + \frac{104698244219447}{31250000000000}} \]

      13. lower-fma.f64 95.3

          \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right) \cdot y}{\color{blue}{\mathsf{fma}\left(z + 6.012459259764103, z, 3.350343815022304\right)}} \]

      14. lift-+.f64 N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\mathsf{fma}\left(\color{blue}{z + \frac{6012459259764103}{1000000000000000}}, z, \frac{104698244219447}{31250000000000}\right)} \]

      15. +-commutative N/A

          \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\mathsf{fma}\left(\color{blue}{\frac{6012459259764103}{1000000000000000} + z}, z, \frac{104698244219447}{31250000000000}\right)} \]

      16. lower-+.f64 95.3

          \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right) \cdot y}{\mathsf{fma}\left(\color{blue}{6.012459259764103 + z}, z, 3.350343815022304\right)} \]

   4. Applied rewrites 95.3%

      \[\leadsto x + \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right) \cdot y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot y}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot y}}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)} \]

      3. associate-*r/ N/A

         \[\leadsto x + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot \frac{y}{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}} \]

      4. clear-num N/A

         \[\leadsto x + \mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right) \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}{y}}} \]

      5. un-div-inv N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right)}{\frac{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}{y}}} \]

      6. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right), z, \frac{11167812716741}{40000000000000}\right)}{\frac{\mathsf{fma}\left(\frac{6012459259764103}{1000000000000000} + z, z, \frac{104698244219447}{31250000000000}\right)}{y}}} \]

      7. lower-/.f64 98.9

         \[\leadsto x + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\color{blue}{\frac{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}{y}}} \]

   6. Applied rewrites 98.9%

      \[\leadsto x + \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\frac{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}{y}}} \]

   if 1.0000000000000001e302 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))

   1. Initial program 0.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 99.7

         \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]

   5. Applied rewrites 99.7%

      \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(0.279195317918525 + \left(0.4917317610505968 + 0.0692910599291889 \cdot z\right) \cdot z\right) \cdot y}{3.350343815022304 + \left(6.012459259764103 + z\right) \cdot z} \leq 10^{+302}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\frac{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}{y}} + x\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(0.279195317918525 + \left(0.4917317610505968 + 0.0692910599291889 \cdot z\right) \cdot z\right) \cdot y}{3.350343815022304 + \left(6.012459259764103 + z\right) \cdot z} \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        (+
         0.279195317918525
         (* (+ 0.4917317610505968 (* 0.0692910599291889 z)) z))
        y)
       (+ 3.350343815022304 (* (+ 6.012459259764103 z) z)))
      1e+302)
   (fma
    (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
    (/ y (fma (+ 6.012459259764103 z) z 3.350343815022304))
    x)
   (+ (* 0.0692910599291889 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((((0.279195317918525 + ((0.4917317610505968 + (0.0692910599291889 * z)) * z)) * y) / (3.350343815022304 + ((6.012459259764103 + z) * z))) <= 1e+302) {
		tmp = fma(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525), (y / fma((6.012459259764103 + z), z, 3.350343815022304)), x);
	} else {
		tmp = (0.0692910599291889 * y) + x;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(0.279195317918525 + Float64(Float64(0.4917317610505968 + Float64(0.0692910599291889 * z)) * z)) * y) / Float64(3.350343815022304 + Float64(Float64(6.012459259764103 + z) * z))) <= 1e+302)
		tmp = fma(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525), Float64(y / fma(Float64(6.012459259764103 + z), z, 3.350343815022304)), x);
	else
		tmp = Float64(Float64(0.0692910599291889 * y) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(0.279195317918525 + N[(N[(0.4917317610505968 + N[(0.0692910599291889 * z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(3.350343815022304 + N[(N[(6.012459259764103 + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+302], N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] * N[(y / N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(0.0692910599291889 * y), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 1.0000000000000001e302

   1. Initial program 95.3%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot y}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} + x \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right) \cdot \frac{y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}, \frac{y}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}, x\right)} \]

   4. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, x\right)} \]

   if 1.0000000000000001e302 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))

   1. Initial program 0.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 99.7

         \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]

   5. Applied rewrites 99.7%

      \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(0.279195317918525 + \left(0.4917317610505968 + 0.0692910599291889 \cdot z\right) \cdot z\right) \cdot y}{3.350343815022304 + \left(6.012459259764103 + z\right) \cdot z} \leq 10^{+302}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{elif}\;z \leq 5:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, -0.00277777777751721, \mathsf{fma}\left(y, 0.08333333333333323, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)
   (if (<= z 5.0)
     (fma (* z y) -0.00277777777751721 (fma y 0.08333333333333323 x))
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	} else if (z <= 5.0) {
		tmp = fma((z * y), -0.00277777777751721, fma(y, 0.08333333333333323, x));
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x);
	elseif (z <= 5.0)
		tmp = fma(Float64(z * y), -0.00277777777751721, fma(y, 0.08333333333333323, x));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5.0], N[(N[(z * y), $MachinePrecision] * -0.00277777777751721 + N[(y * 0.08333333333333323 + x), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 28.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\left(x + \left(\frac{692910599291889}{10000000000000000} \cdot y + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right)\right) - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(\left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{307332350656623}{625000000000000} \cdot \frac{y}{z}\right)} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z} \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \left(\frac{307332350656623}{625000000000000} \cdot \frac{y}{z} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000} \cdot \frac{y}{z}\right)} \]

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

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \color{blue}{\frac{y}{z} \cdot \left(\frac{307332350656623}{625000000000000} - \frac{4166096748901211929300981260567}{10000000000000000000000000000000}\right)} \]

      4. metadata-eval N/A

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{y}{z} \cdot \color{blue}{\frac{751220861604756070699018739433}{10000000000000000000000000000000}} \]

      5. metadata-eval N/A

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{y}{z} \cdot \color{blue}{\frac{\frac{-751220861604756070699018739433}{10000000000000000000000000000000}}{-1}} \]

      6. metadata-eval N/A

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{y}{z} \cdot \frac{\color{blue}{\frac{-307332350656623}{625000000000000} - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}}}{-1} \]

      7. times-frac N/A

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \color{blue}{\frac{y \cdot \left(\frac{-307332350656623}{625000000000000} - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000}\right)}{z \cdot -1}} \]

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

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{\color{blue}{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}}{z \cdot -1} \]

      9. *-commutative N/A

         \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{\color{blue}{-1 \cdot z}} \]

      10. mul-1-neg N/A

          \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      11. distribute-neg-frac2 N/A

          \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}\right)\right)} \]

      12. mul-1-neg N/A

          \[\leadsto \left(x + \frac{692910599291889}{10000000000000000} \cdot y\right) + \color{blue}{-1 \cdot \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}} \]

      13. associate-+r+ N/A

          \[\leadsto \color{blue}{x + \left(\frac{692910599291889}{10000000000000000} \cdot y + -1 \cdot \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z}\right)} \]

      14. +-commutative N/A

          \[\leadsto x + \color{blue}{\left(-1 \cdot \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} + \frac{692910599291889}{10000000000000000} \cdot y\right)} \]

      15. +-commutative N/A

          \[\leadsto \color{blue}{\left(-1 \cdot \frac{\frac{-307332350656623}{625000000000000} \cdot y - \frac{-4166096748901211929300981260567}{10000000000000000000000000000000} \cdot y}{z} + \frac{692910599291889}{10000000000000000} \cdot y\right) + x} \]

   5. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)} \]

   if -5.4000000000000004 < z < 5

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}^{3}}{x \cdot x + \left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} - x \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} - x \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}{{x}^{3} + {\left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} - x \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}{{x}^{3} + {\left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}^{3}}}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, x\right)}}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + \color{blue}{\left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right) \cdot z} \]

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

         \[\leadsto \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + \color{blue}{\left(y \cdot \left(\frac{307332350656623}{2093964884388940} - \frac{1678650474502018223880473708075}{11224803678858206361900017468416}\right)\right)} \cdot z \]

      4. metadata-eval N/A

         \[\leadsto \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + \left(y \cdot \color{blue}{\frac{-155900051080628738716045985239}{56124018394291031809500087342080}}\right) \cdot z \]

      5. *-commutative N/A

         \[\leadsto \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + \color{blue}{\left(\frac{-155900051080628738716045985239}{56124018394291031809500087342080} \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + \color{blue}{\frac{-155900051080628738716045985239}{56124018394291031809500087342080} \cdot \left(y \cdot z\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-155900051080628738716045985239}{56124018394291031809500087342080} \cdot \left(y \cdot z\right) + \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot \frac{-155900051080628738716045985239}{56124018394291031809500087342080}} + \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}, x + \frac{279195317918525}{3350343815022304} \cdot y\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}, x + \frac{279195317918525}{3350343815022304} \cdot y\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}, x + \frac{279195317918525}{3350343815022304} \cdot y\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}, \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y + x}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}, \color{blue}{y \cdot \frac{279195317918525}{3350343815022304}} + x\right) \]

      14. lower-fma.f64 99.1

          \[\leadsto \mathsf{fma}\left(z \cdot y, -0.00277777777751721, \color{blue}{\mathsf{fma}\left(y, 0.08333333333333323, x\right)}\right) \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, -0.00277777777751721, \mathsf{fma}\left(y, 0.08333333333333323, x\right)\right)} \]

   if 5 < z

   1. Initial program 49.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;0.0692910599291889 \cdot y + x\\ \mathbf{elif}\;z \leq 5:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, -0.00277777777751721, \mathsf{fma}\left(y, 0.08333333333333323, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (+ (* 0.0692910599291889 y) x)
   (if (<= z 5.0)
     (fma (* z y) -0.00277777777751721 (fma y 0.08333333333333323 x))
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = (0.0692910599291889 * y) + x;
	} else if (z <= 5.0) {
		tmp = fma((z * y), -0.00277777777751721, fma(y, 0.08333333333333323, x));
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = Float64(Float64(0.0692910599291889 * y) + x);
	elseif (z <= 5.0)
		tmp = fma(Float64(z * y), -0.00277777777751721, fma(y, 0.08333333333333323, x));
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(N[(0.0692910599291889 * y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5.0], N[(N[(z * y), $MachinePrecision] * -0.00277777777751721 + N[(y * 0.08333333333333323 + x), $MachinePrecision]), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 28.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 98.4

         \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]

   5. Applied rewrites 98.4%

      \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]

   if -5.4000000000000004 < z < 5

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}} \]

      2. flip3-+ N/A

         \[\leadsto \color{blue}{\frac{{x}^{3} + {\left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}^{3}}{x \cdot x + \left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} - x \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} - x \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}{{x}^{3} + {\left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}^{3}}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} - x \cdot \frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}{{x}^{3} + {\left(\frac{y \cdot \left(\left(z \cdot \frac{692910599291889}{10000000000000000} + \frac{307332350656623}{625000000000000}\right) \cdot z + \frac{11167812716741}{40000000000000}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}}\right)}^{3}}}} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, x\right)}}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + \color{blue}{\left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right) \cdot z} \]

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

         \[\leadsto \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + \color{blue}{\left(y \cdot \left(\frac{307332350656623}{2093964884388940} - \frac{1678650474502018223880473708075}{11224803678858206361900017468416}\right)\right)} \cdot z \]

      4. metadata-eval N/A

         \[\leadsto \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + \left(y \cdot \color{blue}{\frac{-155900051080628738716045985239}{56124018394291031809500087342080}}\right) \cdot z \]

      5. *-commutative N/A

         \[\leadsto \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + \color{blue}{\left(\frac{-155900051080628738716045985239}{56124018394291031809500087342080} \cdot y\right)} \cdot z \]

      6. associate-*r* N/A

         \[\leadsto \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) + \color{blue}{\frac{-155900051080628738716045985239}{56124018394291031809500087342080} \cdot \left(y \cdot z\right)} \]

      7. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-155900051080628738716045985239}{56124018394291031809500087342080} \cdot \left(y \cdot z\right) + \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot \frac{-155900051080628738716045985239}{56124018394291031809500087342080}} + \left(x + \frac{279195317918525}{3350343815022304} \cdot y\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}, x + \frac{279195317918525}{3350343815022304} \cdot y\right)} \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}, x + \frac{279195317918525}{3350343815022304} \cdot y\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}, x + \frac{279195317918525}{3350343815022304} \cdot y\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}, \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y + x}\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y, \frac{-155900051080628738716045985239}{56124018394291031809500087342080}, \color{blue}{y \cdot \frac{279195317918525}{3350343815022304}} + x\right) \]

      14. lower-fma.f64 99.1

          \[\leadsto \mathsf{fma}\left(z \cdot y, -0.00277777777751721, \color{blue}{\mathsf{fma}\left(y, 0.08333333333333323, x\right)}\right) \]

   7. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, -0.00277777777751721, \mathsf{fma}\left(y, 0.08333333333333323, x\right)\right)} \]

   if 5 < z

   1. Initial program 49.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;0.0692910599291889 \cdot y + x\\ \mathbf{elif}\;z \leq 5:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y, -0.00277777777751721, \mathsf{fma}\left(y, 0.08333333333333323, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;0.0692910599291889 \cdot y + x\\ \mathbf{elif}\;z \leq 5:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (+ (* 0.0692910599291889 y) x)
   (if (<= z 5.0)
     (fma y (fma -0.00277777777751721 z 0.08333333333333323) x)
     (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = (0.0692910599291889 * y) + x;
	} else if (z <= 5.0) {
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = Float64(Float64(0.0692910599291889 * y) + x);
	elseif (z <= 5.0)
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(N[(0.0692910599291889 * y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5.0], N[(y * N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 28.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 98.4

         \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]

   5. Applied rewrites 98.4%

      \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]

   if -5.4000000000000004 < z < 5

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{279195317918525}{3350343815022304} \cdot y + z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right)\right) + x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot \left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right) + \frac{279195317918525}{3350343815022304} \cdot y\right)} + x \]

      3. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\frac{307332350656623}{2093964884388940} \cdot y - \frac{1678650474502018223880473708075}{11224803678858206361900017468416} \cdot y\right) \cdot z} + \frac{279195317918525}{3350343815022304} \cdot y\right) + x \]

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

         \[\leadsto \left(\color{blue}{\left(y \cdot \left(\frac{307332350656623}{2093964884388940} - \frac{1678650474502018223880473708075}{11224803678858206361900017468416}\right)\right)} \cdot z + \frac{279195317918525}{3350343815022304} \cdot y\right) + x \]

      5. associate-*l* N/A

         \[\leadsto \left(\color{blue}{y \cdot \left(\left(\frac{307332350656623}{2093964884388940} - \frac{1678650474502018223880473708075}{11224803678858206361900017468416}\right) \cdot z\right)} + \frac{279195317918525}{3350343815022304} \cdot y\right) + x \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot \left(\left(\frac{307332350656623}{2093964884388940} - \frac{1678650474502018223880473708075}{11224803678858206361900017468416}\right) \cdot z\right) + \color{blue}{y \cdot \frac{279195317918525}{3350343815022304}}\right) + x \]

      7. distribute-lft-out N/A

         \[\leadsto \color{blue}{y \cdot \left(\left(\frac{307332350656623}{2093964884388940} - \frac{1678650474502018223880473708075}{11224803678858206361900017468416}\right) \cdot z + \frac{279195317918525}{3350343815022304}\right)} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(\frac{307332350656623}{2093964884388940} - \frac{1678650474502018223880473708075}{11224803678858206361900017468416}\right) \cdot z + \frac{279195317918525}{3350343815022304}, x\right)} \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(\frac{307332350656623}{2093964884388940} - \frac{1678650474502018223880473708075}{11224803678858206361900017468416}, z, \frac{279195317918525}{3350343815022304}\right)}, x\right) \]

      10. metadata-eval 99.1

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(\color{blue}{-0.00277777777751721}, z, 0.08333333333333323\right), x\right) \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)} \]

   if 5 < z

   1. Initial program 49.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;0.0692910599291889 \cdot y + x\\ \mathbf{elif}\;z \leq 5:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;0.0692910599291889 \cdot y + x\\ \mathbf{elif}\;z \leq 6.2:\\ \;\;\;\;\mathsf{fma}\left(y, 0.08333333333333323, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (+ (* 0.0692910599291889 y) x)
   (if (<= z 6.2) (fma y 0.08333333333333323 x) (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = (0.0692910599291889 * y) + x;
	} else if (z <= 6.2) {
		tmp = fma(y, 0.08333333333333323, x);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = Float64(Float64(0.0692910599291889 * y) + x);
	elseif (z <= 6.2)
		tmp = fma(y, 0.08333333333333323, x);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(N[(0.0692910599291889 * y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 6.2], N[(y * 0.08333333333333323 + x), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 28.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 98.4

         \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]

   5. Applied rewrites 98.4%

      \[\leadsto x + \color{blue}{0.0692910599291889 \cdot y} \]

   if -5.4000000000000004 < z < 6.20000000000000018

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{279195317918525}{3350343815022304} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{279195317918525}{3350343815022304}} + x \]

      3. lower-fma.f64 98.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.08333333333333323, x\right)} \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.08333333333333323, x\right)} \]

   if 6.20000000000000018 < z

   1. Initial program 49.0%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;0.0692910599291889 \cdot y + x\\ \mathbf{elif}\;z \leq 6.2:\\ \;\;\;\;\mathsf{fma}\left(y, 0.08333333333333323, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 6.2:\\ \;\;\;\;\mathsf{fma}\left(y, 0.08333333333333323, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma 0.0692910599291889 y x)
   (if (<= z 6.2) (fma y 0.08333333333333323 x) (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 6.2) {
		tmp = fma(y, 0.08333333333333323, x);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 6.2)
		tmp = fma(y, 0.08333333333333323, x);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 6.2], N[(y * 0.08333333333333323 + x), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004 or 6.20000000000000018 < z

   1. Initial program 39.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   if -5.4000000000000004 < z < 6.20000000000000018

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{279195317918525}{3350343815022304} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{279195317918525}{3350343815022304}} + x \]

      3. lower-fma.f64 98.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.08333333333333323, x\right)} \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.08333333333333323, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 49.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;z \leq 116000:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.5)
   (* 0.0692910599291889 y)
   (if (<= z 116000.0) (* 0.08333333333333323 y) (* 0.0692910599291889 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.5) {
		tmp = 0.0692910599291889 * y;
	} else if (z <= 116000.0) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-6.5d0)) then
        tmp = 0.0692910599291889d0 * y
    else if (z <= 116000.0d0) then
        tmp = 0.08333333333333323d0 * y
    else
        tmp = 0.0692910599291889d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.5) {
		tmp = 0.0692910599291889 * y;
	} else if (z <= 116000.0) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -6.5:
		tmp = 0.0692910599291889 * y
	elif z <= 116000.0:
		tmp = 0.08333333333333323 * y
	else:
		tmp = 0.0692910599291889 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.5)
		tmp = Float64(0.0692910599291889 * y);
	elseif (z <= 116000.0)
		tmp = Float64(0.08333333333333323 * y);
	else
		tmp = Float64(0.0692910599291889 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -6.5)
		tmp = 0.0692910599291889 * y;
	elseif (z <= 116000.0)
		tmp = 0.08333333333333323 * y;
	else
		tmp = 0.0692910599291889 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.5], N[(0.0692910599291889 * y), $MachinePrecision], If[LessEqual[z, 116000.0], N[(0.08333333333333323 * y), $MachinePrecision], N[(0.0692910599291889 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5 or 116000 < z

   1. Initial program 39.2%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + \frac{692910599291889}{10000000000000000} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

      2. lower-fma.f64 99.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{692910599291889}{10000000000000000} \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.1%

      \[\leadsto 0.0692910599291889 \cdot \color{blue}{y} \]

   if -6.5 < z < 116000

   1. Initial program 99.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{279195317918525}{3350343815022304} \cdot y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{279195317918525}{3350343815022304} \cdot y + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{y \cdot \frac{279195317918525}{3350343815022304}} + x \]

      3. lower-fma.f64 98.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.08333333333333323, x\right)} \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.08333333333333323, x\right)} \]

   6. Taylor expanded in y around inf

      \[\leadsto \frac{279195317918525}{3350343815022304} \cdot \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.0%

      \[\leadsto y \cdot \color{blue}{0.08333333333333323} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;z \leq 116000:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.9% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ 0.0692910599291889 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.0692910599291889 y))

C

double code(double x, double y, double z) {
	return 0.0692910599291889 * y;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.0692910599291889d0 * y
end function

Java

public static double code(double x, double y, double z) {
	return 0.0692910599291889 * y;
}

Python

def code(x, y, z):
	return 0.0692910599291889 * y

Julia

function code(x, y, z)
	return Float64(0.0692910599291889 * y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0692910599291889 * y;
end

Wolfram

code[x_, y_, z_] := N[(0.0692910599291889 * y), $MachinePrecision]

Derivation

1. Initial program 72.0%

   \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x + \frac{692910599291889}{10000000000000000} \cdot y} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{692910599291889}{10000000000000000} \cdot y + x} \]

   2. lower-fma.f64 75.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

5. Applied rewrites 75.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]

6. Taylor expanded in y around inf

   \[\leadsto \frac{692910599291889}{10000000000000000} \cdot \color{blue}{y} \]
7. Step-by-step derivation

8. Applied rewrites 28.0%

   \[\leadsto 0.0692910599291889 \cdot \color{blue}{y} \]
9. Add Preprocessing

Developer Target 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{0.07512208616047561}{z} + 0.0692910599291889\right) \cdot y - \left(\frac{0.40462203869992125 \cdot y}{z \cdot z} - x\right)\\ \mathbf{if}\;z < -8120153.652456675:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z < 6.576118972787377 \cdot 10^{+20}:\\ \;\;\;\;x + \left(y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)\right) \cdot \frac{1}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (-
          (* (+ (/ 0.07512208616047561 z) 0.0692910599291889) y)
          (- (/ (* 0.40462203869992125 y) (* z z)) x))))
   (if (< z -8120153.652456675)
     t_0
     (if (< z 6.576118972787377e+20)
       (+
        x
        (*
         (*
          y
          (+
           (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
           0.279195317918525))
         (/ 1.0 (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) - (((0.40462203869992125 * y) / (z * z)) - x);
	double tmp;
	if (z < -8120153.652456675) {
		tmp = t_0;
	} else if (z < 6.576118972787377e+20) {
		tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * (1.0 / (((z + 6.012459259764103) * z) + 3.350343815022304)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((0.07512208616047561d0 / z) + 0.0692910599291889d0) * y) - (((0.40462203869992125d0 * y) / (z * z)) - x)
    if (z < (-8120153.652456675d0)) then
        tmp = t_0
    else if (z < 6.576118972787377d+20) then
        tmp = x + ((y * ((((z * 0.0692910599291889d0) + 0.4917317610505968d0) * z) + 0.279195317918525d0)) * (1.0d0 / (((z + 6.012459259764103d0) * z) + 3.350343815022304d0)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) - (((0.40462203869992125 * y) / (z * z)) - x);
	double tmp;
	if (z < -8120153.652456675) {
		tmp = t_0;
	} else if (z < 6.576118972787377e+20) {
		tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * (1.0 / (((z + 6.012459259764103) * z) + 3.350343815022304)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) - (((0.40462203869992125 * y) / (z * z)) - x)
	tmp = 0
	if z < -8120153.652456675:
		tmp = t_0
	elif z < 6.576118972787377e+20:
		tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * (1.0 / (((z + 6.012459259764103) * z) + 3.350343815022304)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(0.07512208616047561 / z) + 0.0692910599291889) * y) - Float64(Float64(Float64(0.40462203869992125 * y) / Float64(z * z)) - x))
	tmp = 0.0
	if (z < -8120153.652456675)
		tmp = t_0;
	elseif (z < 6.576118972787377e+20)
		tmp = Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * Float64(1.0 / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((0.07512208616047561 / z) + 0.0692910599291889) * y) - (((0.40462203869992125 * y) / (z * z)) - x);
	tmp = 0.0;
	if (z < -8120153.652456675)
		tmp = t_0;
	elseif (z < 6.576118972787377e+20)
		tmp = x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) * (1.0 / (((z + 6.012459259764103) * z) + 3.350343815022304)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(0.07512208616047561 / z), $MachinePrecision] + 0.0692910599291889), $MachinePrecision] * y), $MachinePrecision] - N[(N[(N[(0.40462203869992125 * y), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -8120153.652456675], t$95$0, If[Less[z, 6.576118972787377e+20], N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024270 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -324806146098267/40000000) (- (* (+ (/ 7512208616047561/100000000000000000 z) 692910599291889/10000000000000000) y) (- (/ (* 323697630959937/800000000000000 y) (* z z)) x)) (if (< z 657611897278737700000) (+ x (* (* y (+ (* (+ (* z 692910599291889/10000000000000000) 307332350656623/625000000000000) z) 11167812716741/40000000000000)) (/ 1 (+ (* (+ z 6012459259764103/1000000000000000) z) 104698244219447/31250000000000)))) (- (* (+ (/ 7512208616047561/100000000000000000 z) 692910599291889/10000000000000000) y) (- (/ (* 323697630959937/800000000000000 y) (* z z)) x)))))

  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))