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

Percentage Accurate: 69.1% → 99.3%

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: 69.1% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \lor \neg \left(z \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889 - \frac{\frac{0.4046220386999212}{z} - 0.07512208616047561}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005951669793454025, z, 0.0007936505811533442\right) \cdot z - 0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2) (not (<= z 2.7e-5)))
   (fma
    (-
     0.0692910599291889
     (/ (- (/ 0.4046220386999212 z) 0.07512208616047561) z))
    y
    x)
   (fma
    (fma
     (-
      (* (fma -0.0005951669793454025 z 0.0007936505811533442) z)
      0.00277777777751721)
     z
     0.08333333333333323)
    y
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2) || !(z <= 2.7e-5)) {
		tmp = fma((0.0692910599291889 - (((0.4046220386999212 / z) - 0.07512208616047561) / z)), y, x);
	} else {
		tmp = fma(fma(((fma(-0.0005951669793454025, z, 0.0007936505811533442) * z) - 0.00277777777751721), z, 0.08333333333333323), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2) || !(z <= 2.7e-5))
		tmp = fma(Float64(0.0692910599291889 - Float64(Float64(Float64(0.4046220386999212 / z) - 0.07512208616047561) / z)), y, x);
	else
		tmp = fma(fma(Float64(Float64(fma(-0.0005951669793454025, z, 0.0007936505811533442) * z) - 0.00277777777751721), z, 0.08333333333333323), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.2], N[Not[LessEqual[z, 2.7e-5]], $MachinePrecision]], N[(N[(0.0692910599291889 - N[(N[(N[(0.4046220386999212 / z), $MachinePrecision] - 0.07512208616047561), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(N[(N[(-0.0005951669793454025 * z + 0.0007936505811533442), $MachinePrecision] * z), $MachinePrecision] - 0.00277777777751721), $MachinePrecision] * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000018 or 2.6999999999999999e-5 < z

   1. Initial program 43.3%

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

   3. Taylor expanded in z around inf

      \[\leadsto x + \frac{y \cdot \color{blue}{\left({z}^{2} \cdot \left(\frac{692910599291889}{10000000000000000} + \left(\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)\right)\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(\frac{692910599291889}{10000000000000000} + \left(\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      2. associate-*l* N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{692910599291889}{10000000000000000} + \left(\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)\right)\right)\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{692910599291889}{10000000000000000} + \left(\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)\right)\right)\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      4. associate-+r+ N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(z \cdot \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      5. distribute-rgt-in N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) \cdot z + \left(\frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right) \cdot z\right)}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      6. associate-*l* N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) \cdot z + \color{blue}{\frac{307332350656623}{625000000000000} \cdot \left(\frac{1}{z} \cdot z\right)}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      7. lft-mult-inverse N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) \cdot z + \frac{307332350656623}{625000000000000} \cdot \color{blue}{1}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      8. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) \cdot z + \color{blue}{\frac{307332350656623}{625000000000000}}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      9. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}, z, \frac{307332350656623}{625000000000000}\right)}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      10. +-commutative N/A

          \[\leadsto x + \frac{y \cdot \left(z \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{692910599291889}{10000000000000000}}, z, \frac{307332350656623}{625000000000000}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      11. lower-+.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(z \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{692910599291889}{10000000000000000}}, z, \frac{307332350656623}{625000000000000}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      12. lower-/.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(z \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}} + \frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      13. unpow2 N/A

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

      14. lower-*.f64 43.2

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

   5. Applied rewrites 43.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 50.1%

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

   8. Taylor expanded in z around -inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} + -1 \cdot \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}}, y, x\right) \]
   9. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} - \left(\mathsf{neg}\left(-1\right)\right) \cdot \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}}, y, x\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} - \color{blue}{1} \cdot \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}, y, x\right) \]

      3. *-lft-identity N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} - \color{blue}{\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}}, y, x\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000} - \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}}, y, x\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} - \color{blue}{\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}}, y, x\right) \]

      6. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} - \frac{\color{blue}{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}}{z}, y, x\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} - \frac{\color{blue}{\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot 1}{z}} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}, y, x\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{692910599291889}{10000000000000000} - \frac{\frac{\color{blue}{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}}}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}, y, x\right) \]

      9. lower-/.f64 99.0

         \[\leadsto \mathsf{fma}\left(0.0692910599291889 - \frac{\color{blue}{\frac{0.4046220386999212}{z}} - 0.07512208616047561}{z}, y, x\right) \]

   10. Applied rewrites 99.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{0.0692910599291889 - \frac{\frac{0.4046220386999212}{z} - 0.07512208616047561}{z}}, y, x\right) \]

   if -5.20000000000000018 < z < 2.6999999999999999e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto x + \frac{y \cdot \color{blue}{\left({z}^{2} \cdot \left(\frac{692910599291889}{10000000000000000} + \left(\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)\right)\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(\frac{692910599291889}{10000000000000000} + \left(\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      2. associate-*l* N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{692910599291889}{10000000000000000} + \left(\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)\right)\right)\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{692910599291889}{10000000000000000} + \left(\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)\right)\right)\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      4. associate-+r+ N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(z \cdot \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      5. distribute-rgt-in N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) \cdot z + \left(\frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right) \cdot z\right)}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      6. associate-*l* N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) \cdot z + \color{blue}{\frac{307332350656623}{625000000000000} \cdot \left(\frac{1}{z} \cdot z\right)}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      7. lft-mult-inverse N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) \cdot z + \frac{307332350656623}{625000000000000} \cdot \color{blue}{1}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      8. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) \cdot z + \color{blue}{\frac{307332350656623}{625000000000000}}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      9. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}, z, \frac{307332350656623}{625000000000000}\right)}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      10. +-commutative N/A

          \[\leadsto x + \frac{y \cdot \left(z \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{692910599291889}{10000000000000000}}, z, \frac{307332350656623}{625000000000000}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      11. lower-+.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(z \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{692910599291889}{10000000000000000}}, z, \frac{307332350656623}{625000000000000}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      12. lower-/.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(z \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}} + \frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      13. unpow2 N/A

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

      14. lower-*.f64 45.6

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

   5. Applied rewrites 45.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 45.6%

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

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{279195317918525}{3350343815022304} + z \cdot \left(z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, y, x\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right) + \frac{279195317918525}{3350343815022304}}, y, x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right) \cdot z} + \frac{279195317918525}{3350343815022304}, y, x\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}, z, \frac{279195317918525}{3350343815022304}\right)}, y, x\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}}, z, \frac{279195317918525}{3350343815022304}\right), y, x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) \cdot z} - \frac{155900051080628738716045985239}{56124018394291031809500087342080}, z, \frac{279195317918525}{3350343815022304}\right), y, x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) \cdot z} - \frac{155900051080628738716045985239}{56124018394291031809500087342080}, z, \frac{279195317918525}{3350343815022304}\right), y, x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z + \frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320}\right)} \cdot z - \frac{155900051080628738716045985239}{56124018394291031809500087342080}, z, \frac{279195317918525}{3350343815022304}\right), y, x\right) \]

      8. lower-fma.f64 99.1

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.0005951669793454025, z, 0.0007936505811533442\right)} \cdot z - 0.00277777777751721, z, 0.08333333333333323\right), y, x\right) \]

   10. Applied rewrites 99.1%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0005951669793454025, z, 0.0007936505811533442\right) \cdot z - 0.00277777777751721, z, 0.08333333333333323\right)}, y, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \lor \neg \left(z \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889 - \frac{\frac{0.4046220386999212}{z} - 0.07512208616047561}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005951669793454025, z, 0.0007936505811533442\right) \cdot z - 0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq -2 \cdot 10^{+141}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 -2e+141)))
     (fma 0.0692910599291889 y x)
     (* 0.08333333333333323 y))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= -2e+141)) {
		tmp = fma(0.0692910599291889, y, x);
	} else {
		tmp = 0.08333333333333323 * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= -2e+141))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = Float64(0.08333333333333323 * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = 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]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, -2e+141]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(0.08333333333333323 * y), $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 -2.00000000000000003e141 < (/.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 67.2%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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 87.0

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

   5. Applied rewrites 87.0%

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

   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))) < -2.00000000000000003e141

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 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. lower-fma.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.1%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq -\infty \lor \neg \left(\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq -2 \cdot 10^{+141}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       x
       (/
        (*
         y
         (+
          (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
          0.279195317918525))
        (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
      2e+305)
   (fma
    (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
    (/ y (fma (+ 6.012459259764103 z) z 3.350343815022304))
    x)
   (fma 0.0692910599291889 y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 2e+305) {
		tmp = fma(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525), (y / fma((6.012459259764103 + z), z, 3.350343815022304)), x);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (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))) <= 2e+305)
		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 = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[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], 2e+305], 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[(0.0692910599291889 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.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)))) < 1.9999999999999999e305

   1. Initial program 96.7%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. +-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 98.6%

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

   if 1.9999999999999999e305 < (+.f64 x (/.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 0.4%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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 99.6

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \lor \neg \left(z \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right)}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005951669793454025, z, 0.0007936505811533442\right) \cdot z - 0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2) (not (<= z 2.7e-5)))
   (fma y (/ (fma 0.0692910599291889 z 0.07512208616047561) z) x)
   (fma
    (fma
     (-
      (* (fma -0.0005951669793454025 z 0.0007936505811533442) z)
      0.00277777777751721)
     z
     0.08333333333333323)
    y
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2) || !(z <= 2.7e-5)) {
		tmp = fma(y, (fma(0.0692910599291889, z, 0.07512208616047561) / z), x);
	} else {
		tmp = fma(fma(((fma(-0.0005951669793454025, z, 0.0007936505811533442) * z) - 0.00277777777751721), z, 0.08333333333333323), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2) || !(z <= 2.7e-5))
		tmp = fma(y, Float64(fma(0.0692910599291889, z, 0.07512208616047561) / z), x);
	else
		tmp = fma(fma(Float64(Float64(fma(-0.0005951669793454025, z, 0.0007936505811533442) * z) - 0.00277777777751721), z, 0.08333333333333323), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.2], N[Not[LessEqual[z, 2.7e-5]], $MachinePrecision]], N[(y * N[(N[(0.0692910599291889 * z + 0.07512208616047561), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(N[(-0.0005951669793454025 * z + 0.0007936505811533442), $MachinePrecision] * z), $MachinePrecision] - 0.00277777777751721), $MachinePrecision] * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000018 or 2.6999999999999999e-5 < z

   1. Initial program 43.3%

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

   3. 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--l+ N/A

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

      2. +-commutative N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. associate-+l- N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

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

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

      13. associate-/l* N/A

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

      14. distribute-lft-out-- N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} + \frac{692910599291889}{10000000000000000} \cdot z}{\color{blue}{z}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.6%

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

   if -5.20000000000000018 < z < 2.6999999999999999e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto x + \frac{y \cdot \color{blue}{\left({z}^{2} \cdot \left(\frac{692910599291889}{10000000000000000} + \left(\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)\right)\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto x + \frac{y \cdot \left(\color{blue}{\left(z \cdot z\right)} \cdot \left(\frac{692910599291889}{10000000000000000} + \left(\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      2. associate-*l* N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{692910599291889}{10000000000000000} + \left(\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)\right)\right)\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      3. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \color{blue}{\left(z \cdot \left(z \cdot \left(\frac{692910599291889}{10000000000000000} + \left(\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)\right)\right)\right)}}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      4. associate-+r+ N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(z \cdot \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) + \frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right)}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      5. distribute-rgt-in N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \color{blue}{\left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) \cdot z + \left(\frac{307332350656623}{625000000000000} \cdot \frac{1}{z}\right) \cdot z\right)}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      6. associate-*l* N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) \cdot z + \color{blue}{\frac{307332350656623}{625000000000000} \cdot \left(\frac{1}{z} \cdot z\right)}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      7. lft-mult-inverse N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) \cdot z + \frac{307332350656623}{625000000000000} \cdot \color{blue}{1}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      8. metadata-eval N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \left(\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}\right) \cdot z + \color{blue}{\frac{307332350656623}{625000000000000}}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      9. lower-fma.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(z \cdot \color{blue}{\mathsf{fma}\left(\frac{692910599291889}{10000000000000000} + \frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}, z, \frac{307332350656623}{625000000000000}\right)}\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      10. +-commutative N/A

          \[\leadsto x + \frac{y \cdot \left(z \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{692910599291889}{10000000000000000}}, z, \frac{307332350656623}{625000000000000}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      11. lower-+.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(z \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}} + \frac{692910599291889}{10000000000000000}}, z, \frac{307332350656623}{625000000000000}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      12. lower-/.f64 N/A

          \[\leadsto x + \frac{y \cdot \left(z \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{11167812716741}{40000000000000}}{{z}^{2}}} + \frac{692910599291889}{10000000000000000}, z, \frac{307332350656623}{625000000000000}\right)\right)}{\left(z + \frac{6012459259764103}{1000000000000000}\right) \cdot z + \frac{104698244219447}{31250000000000}} \]

      13. unpow2 N/A

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

      14. lower-*.f64 45.6

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

   5. Applied rewrites 45.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 45.6%

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

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{279195317918525}{3350343815022304} + z \cdot \left(z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right)}, y, x\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot \left(z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right) + \frac{279195317918525}{3350343815022304}}, y, x\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}\right) \cdot z} + \frac{279195317918525}{3350343815022304}, y, x\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}, z, \frac{279195317918525}{3350343815022304}\right)}, y, x\right) \]

      4. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) - \frac{155900051080628738716045985239}{56124018394291031809500087342080}}, z, \frac{279195317918525}{3350343815022304}\right), y, x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) \cdot z} - \frac{155900051080628738716045985239}{56124018394291031809500087342080}, z, \frac{279195317918525}{3350343815022304}\right), y, x\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320} + \frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z\right) \cdot z} - \frac{155900051080628738716045985239}{56124018394291031809500087342080}, z, \frac{279195317918525}{3350343815022304}\right), y, x\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(\frac{-374943941275717765274452559944207169728571246668095556552487}{629981088144543617699065742275429975113587435159029727787745280} \cdot z + \frac{149233894885562575800992648418763933371314529}{188034757901510979839193143041976607183277752320}\right)} \cdot z - \frac{155900051080628738716045985239}{56124018394291031809500087342080}, z, \frac{279195317918525}{3350343815022304}\right), y, x\right) \]

      8. lower-fma.f64 99.1

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.0005951669793454025, z, 0.0007936505811533442\right)} \cdot z - 0.00277777777751721, z, 0.08333333333333323\right), y, x\right) \]

   10. Applied rewrites 99.1%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0005951669793454025, z, 0.0007936505811533442\right) \cdot z - 0.00277777777751721, z, 0.08333333333333323\right)}, y, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \lor \neg \left(z \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right)}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005951669793454025, z, 0.0007936505811533442\right) \cdot z - 0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \lor \neg \left(z \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right)}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2) (not (<= z 2.7e-5)))
   (fma y (/ (fma 0.0692910599291889 z 0.07512208616047561) z) x)
   (fma y (fma -0.00277777777751721 z 0.08333333333333323) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2) || !(z <= 2.7e-5)) {
		tmp = fma(y, (fma(0.0692910599291889, z, 0.07512208616047561) / z), x);
	} else {
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2) || !(z <= 2.7e-5))
		tmp = fma(y, Float64(fma(0.0692910599291889, z, 0.07512208616047561) / z), x);
	else
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.2], N[Not[LessEqual[z, 2.7e-5]], $MachinePrecision]], N[(y * N[(N[(0.0692910599291889 * z + 0.07512208616047561), $MachinePrecision] / z), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000018 or 2.6999999999999999e-5 < z

   1. Initial program 43.3%

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

   3. 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--l+ N/A

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

      2. +-commutative N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. associate-+l- N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

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

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

      13. associate-/l* N/A

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

      14. distribute-lft-out-- N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, \frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} + \frac{692910599291889}{10000000000000000} \cdot z}{\color{blue}{z}}, x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.6%

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

   if -5.20000000000000018 < z < 2.6999999999999999e-5

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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 98.9

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

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \lor \neg \left(z \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(0.0692910599291889, z, 0.07512208616047561\right)}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \lor \neg \left(z \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2) (not (<= z 2.7e-5)))
   (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)
   (fma y (fma -0.00277777777751721 z 0.08333333333333323) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2) || !(z <= 2.7e-5)) {
		tmp = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	} else {
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2) || !(z <= 2.7e-5))
		tmp = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x);
	else
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.2], N[Not[LessEqual[z, 2.7e-5]], $MachinePrecision]], N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000018 or 2.6999999999999999e-5 < z

   1. Initial program 43.3%

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

   3. 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--l+ N/A

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

      2. +-commutative N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. associate-+l- N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. div-sub N/A

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

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

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

      13. associate-/l* N/A

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

      14. distribute-lft-out-- N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 98.6%

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

   if -5.20000000000000018 < z < 2.6999999999999999e-5

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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 98.9

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

   5. Applied rewrites 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \lor \neg \left(z \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.2)
   (fma 0.0692910599291889 y x)
   (if (<= z 2.7e-5)
     (fma y (fma -0.00277777777751721 z 0.08333333333333323) x)
     (+ x (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.2) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 2.7e-5) {
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.2)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 2.7e-5)
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.2], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 2.7e-5], N[(y * N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.20000000000000018

   1. Initial program 36.6%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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 98.2

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

   5. Applied rewrites 98.2%

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

   if -5.20000000000000018 < z < 2.6999999999999999e-5

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   if 2.6999999999999999e-5 < z

   1. Initial program 51.6%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

Alternative 8: 98.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.2)
   (fma 0.0692910599291889 y x)
   (if (<= z 2.7e-5)
     (fma 0.08333333333333323 y x)
     (+ x (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.2) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 2.7e-5) {
		tmp = fma(0.08333333333333323, y, x);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.2)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 2.7e-5)
		tmp = fma(0.08333333333333323, y, x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.2], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 2.7e-5], N[(0.08333333333333323 * y + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.20000000000000018

   1. Initial program 36.6%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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 98.2

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

   5. Applied rewrites 98.2%

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

   if -5.20000000000000018 < z < 2.6999999999999999e-5

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. lower-fma.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if 2.6999999999999999e-5 < z

   1. Initial program 51.6%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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 97.7

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

   5. Applied rewrites 97.7%

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

Alternative 9: 98.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \lor \neg \left(z \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.2) (not (<= z 2.7e-5)))
   (fma 0.0692910599291889 y x)
   (fma 0.08333333333333323 y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.2) || !(z <= 2.7e-5)) {
		tmp = fma(0.0692910599291889, y, x);
	} else {
		tmp = fma(0.08333333333333323, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.2) || !(z <= 2.7e-5))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = fma(0.08333333333333323, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.2], N[Not[LessEqual[z, 2.7e-5]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(0.08333333333333323 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.20000000000000018 or 2.6999999999999999e-5 < z

   1. Initial program 43.3%

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

   3. 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 98.0

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

   5. Applied rewrites 98.0%

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

   if -5.20000000000000018 < z < 2.6999999999999999e-5

   1. Initial program 99.8%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. lower-fma.f64 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \lor \neg \left(z \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+30} \lor \neg \left(x \leq 7.6 \cdot 10^{-96}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.15e+30) (not (<= x 7.6e-96)))
   (* 1.0 x)
   (* 0.0692910599291889 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.15e+30) || !(x <= 7.6e-96)) {
		tmp = 1.0 * x;
	} 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 ((x <= (-2.15d+30)) .or. (.not. (x <= 7.6d-96))) then
        tmp = 1.0d0 * x
    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 ((x <= -2.15e+30) || !(x <= 7.6e-96)) {
		tmp = 1.0 * x;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.15e+30) or not (x <= 7.6e-96):
		tmp = 1.0 * x
	else:
		tmp = 0.0692910599291889 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.15e+30) || !(x <= 7.6e-96))
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(0.0692910599291889 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.15e+30) || ~((x <= 7.6e-96)))
		tmp = 1.0 * x;
	else
		tmp = 0.0692910599291889 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.15e+30], N[Not[LessEqual[x, 7.6e-96]], $MachinePrecision]], N[(1.0 * x), $MachinePrecision], N[(0.0692910599291889 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.15e30 or 7.6000000000000001e-96 < x

   1. Initial program 69.8%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(1 + \frac{692910599291889}{10000000000000000} \cdot \frac{y}{x}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 91.3%

      \[\leadsto \mathsf{fma}\left(\frac{y}{x}, 0.0692910599291889, 1\right) \cdot \color{blue}{x} \]

   9. Taylor expanded in x around inf

      \[\leadsto 1 \cdot x \]
   10. Step-by-step derivation

   11. Applied rewrites 79.8%

       \[\leadsto 1 \cdot x \]

   if -2.15e30 < x < 7.6000000000000001e-96

   1. Initial program 70.2%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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 69.7

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.3%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{+30} \lor \neg \left(x \leq 7.6 \cdot 10^{-96}\right):\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 31.0% 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 70.0%

   \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
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 82.6

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

5. Applied rewrites 82.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 28.7%

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

Developer Target 1: 99.4% 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 2024338 
(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))))