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

Percentage Accurate: 68.8% → 99.6%

Time: 14.5s

Alternatives: 14

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: 68.8% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{y}{\frac{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{14.431876219268936 + \frac{-15.646356830292042}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        y
        (+
         (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
         0.279195317918525))
       (+ (* z (+ z 6.012459259764103)) 3.350343815022304))
      2e+307)
   (+
    x
    (/
     y
     (/
      (fma z (+ z 6.012459259764103) 3.350343815022304)
      (fma
       z
       (fma z 0.0692910599291889 0.4917317610505968)
       0.279195317918525))))
   (+ x (/ y (+ 14.431876219268936 (/ -15.646356830292042 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304)) <= 2e+307) {
		tmp = x + (y / (fma(z, (z + 6.012459259764103), 3.350343815022304) / fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525)));
	} else {
		tmp = x + (y / (14.431876219268936 + (-15.646356830292042 / z)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304)) <= 2e+307)
		tmp = Float64(x + Float64(y / Float64(fma(z, Float64(z + 6.012459259764103), 3.350343815022304) / fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525))));
	else
		tmp = Float64(x + Float64(y / Float64(14.431876219268936 + Float64(-15.646356830292042 / z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision], 2e+307], N[(x + N[(y / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision] + 3.350343815022304), $MachinePrecision] / N[(z * N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y / N[(14.431876219268936 + N[(-15.646356830292042 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 1.99999999999999997e307

   1. Initial program 94.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-lowering-+.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   if 1.99999999999999997e307 < (/.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 1.1%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-lowering-+.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 16.5

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

   4. Applied egg-rr 16.5%

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

   5. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\color{blue}{\frac{10000000000000000}{692910599291889} - \frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot \frac{1}{z}}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot \frac{1}{z}\right)\right)}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot \frac{1}{z}\right)\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot 1}{z}}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2504069538682520235663395798110}{160041699537014921582740396107}}}{z}\right)\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107}\right)}{z}}} \]

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

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107}\right)}{z}}} \]

      7. metadata-eval 99.9

         \[\leadsto x + \frac{y}{14.431876219268936 + \frac{\color{blue}{-15.646356830292042}}{z}} \]

   7. Simplified 99.9%

      \[\leadsto x + \frac{y}{\color{blue}{14.431876219268936 + \frac{-15.646356830292042}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{y}{\frac{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{14.431876219268936 + \frac{-15.646356830292042}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+184}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
            0.279195317918525))
          (+ (* z (+ z 6.012459259764103)) 3.350343815022304))))
   (if (<= t_0 (- INFINITY))
     (* y 0.0692910599291889)
     (if (<= t_0 -1e+184)
       (* y 0.08333333333333323)
       (if (<= t_0 5e+216)
         (fma y 0.0692910599291889 x)
         (if (<= t_0 2e+307)
           (* y 0.08333333333333323)
           (fma y 0.0692910599291889 x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = y * 0.0692910599291889;
	} else if (t_0 <= -1e+184) {
		tmp = y * 0.08333333333333323;
	} else if (t_0 <= 5e+216) {
		tmp = fma(y, 0.0692910599291889, x);
	} else if (t_0 <= 2e+307) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(y * 0.0692910599291889);
	elseif (t_0 <= -1e+184)
		tmp = Float64(y * 0.08333333333333323);
	elseif (t_0 <= 5e+216)
		tmp = fma(y, 0.0692910599291889, x);
	elseif (t_0 <= 2e+307)
		tmp = Float64(y * 0.08333333333333323);
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(y * 0.0692910599291889), $MachinePrecision], If[LessEqual[t$95$0, -1e+184], N[(y * 0.08333333333333323), $MachinePrecision], If[LessEqual[t$95$0, 5e+216], N[(y * 0.0692910599291889 + x), $MachinePrecision], If[LessEqual[t$95$0, 2e+307], N[(y * 0.08333333333333323), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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

   1. Initial program 6.3%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 6.3

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

   5. Simplified 6.3%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

   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))) < -1.00000000000000002e184 or 4.9999999999999998e216 < (/.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))) < 1.99999999999999997e307

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 99.8

         \[\leadsto x + \color{blue}{y \cdot 0.08333333333333323} \]

   5. Simplified 99.8%

      \[\leadsto x + \color{blue}{y \cdot 0.08333333333333323} \]

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 90.9

         \[\leadsto \color{blue}{0.08333333333333323 \cdot y} \]

   8. Simplified 90.9%

      \[\leadsto \color{blue}{0.08333333333333323 \cdot y} \]

   if -1.00000000000000002e184 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 4.9999999999999998e216 or 1.99999999999999997e307 < (/.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 72.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 82.6

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

   5. Simplified 82.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq -\infty:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq -1 \cdot 10^{+184}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 5 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;t\_0 \leq -4 \cdot 10^{-47}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;t\_0 \leq 10^{+111}:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
            0.279195317918525))
          (+ (* z (+ z 6.012459259764103)) 3.350343815022304))))
   (if (<= t_0 (- INFINITY))
     (* y 0.0692910599291889)
     (if (<= t_0 -4e-47)
       (* y 0.08333333333333323)
       (if (<= t_0 1e+111)
         x
         (if (<= t_0 2e+307) (* y 0.08333333333333323) x))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = y * 0.0692910599291889;
	} else if (t_0 <= -4e-47) {
		tmp = y * 0.08333333333333323;
	} else if (t_0 <= 1e+111) {
		tmp = x;
	} else if (t_0 <= 2e+307) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = y * 0.0692910599291889;
	} else if (t_0 <= -4e-47) {
		tmp = y * 0.08333333333333323;
	} else if (t_0 <= 1e+111) {
		tmp = x;
	} else if (t_0 <= 2e+307) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = y * 0.0692910599291889
	elif t_0 <= -4e-47:
		tmp = y * 0.08333333333333323
	elif t_0 <= 1e+111:
		tmp = x
	elif t_0 <= 2e+307:
		tmp = y * 0.08333333333333323
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(y * 0.0692910599291889);
	elseif (t_0 <= -4e-47)
		tmp = Float64(y * 0.08333333333333323);
	elseif (t_0 <= 1e+111)
		tmp = x;
	elseif (t_0 <= 2e+307)
		tmp = Float64(y * 0.08333333333333323);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = y * 0.0692910599291889;
	elseif (t_0 <= -4e-47)
		tmp = y * 0.08333333333333323;
	elseif (t_0 <= 1e+111)
		tmp = x;
	elseif (t_0 <= 2e+307)
		tmp = y * 0.08333333333333323;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(y * 0.0692910599291889), $MachinePrecision], If[LessEqual[t$95$0, -4e-47], N[(y * 0.08333333333333323), $MachinePrecision], If[LessEqual[t$95$0, 1e+111], x, If[LessEqual[t$95$0, 2e+307], N[(y * 0.08333333333333323), $MachinePrecision], x]]]]]

Derivation

1. Split input into 3 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

   1. Initial program 6.3%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

      8. +-commutative N/A

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

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

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

      10. +-commutative N/A

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

      11. +-lowering-+.f64 6.3

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

   5. Simplified 6.3%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 99.2

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

   8. Simplified 99.2%

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

   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))) < -3.9999999999999999e-47 or 9.99999999999999957e110 < (/.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))) < 1.99999999999999997e307

   1. Initial program 99.3%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 86.4

         \[\leadsto x + \color{blue}{y \cdot 0.08333333333333323} \]

   5. Simplified 86.4%

      \[\leadsto x + \color{blue}{y \cdot 0.08333333333333323} \]

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 64.9

         \[\leadsto \color{blue}{0.08333333333333323 \cdot y} \]

   8. Simplified 64.9%

      \[\leadsto \color{blue}{0.08333333333333323 \cdot y} \]

   if -3.9999999999999999e-47 < (/.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))) < 9.99999999999999957e110 or 1.99999999999999997e307 < (/.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 64.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 64.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq -\infty:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq -4 \cdot 10^{-47}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 10^{+111}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{14.431876219268936 + \frac{-15.646356830292042}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        y
        (+
         (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
         0.279195317918525))
       (+ (* z (+ z 6.012459259764103)) 3.350343815022304))
      2e+307)
   (fma
    (fma z (fma z 0.0692910599291889 0.4917317610505968) 0.279195317918525)
    (/ y (fma z (+ z 6.012459259764103) 3.350343815022304))
    x)
   (+ x (/ y (+ 14.431876219268936 (/ -15.646356830292042 z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304)) <= 2e+307) {
		tmp = fma(fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525), (y / fma(z, (z + 6.012459259764103), 3.350343815022304)), x);
	} else {
		tmp = x + (y / (14.431876219268936 + (-15.646356830292042 / z)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304)) <= 2e+307)
		tmp = fma(fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525), Float64(y / fma(z, Float64(z + 6.012459259764103), 3.350343815022304)), x);
	else
		tmp = Float64(x + Float64(y / Float64(14.431876219268936 + Float64(-15.646356830292042 / z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision], 2e+307], N[(N[(z * N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision] + 0.279195317918525), $MachinePrecision] * N[(y / N[(z * N[(z + 6.012459259764103), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(y / N[(14.431876219268936 + N[(-15.646356830292042 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 1.99999999999999997e307

   1. Initial program 94.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. +-lowering-+.f64 98.1

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

   4. Applied egg-rr 98.1%

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

   if 1.99999999999999997e307 < (/.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 1.1%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-lowering-+.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 16.5

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

   4. Applied egg-rr 16.5%

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

   5. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\color{blue}{\frac{10000000000000000}{692910599291889} - \frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot \frac{1}{z}}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot \frac{1}{z}\right)\right)}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot \frac{1}{z}\right)\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot 1}{z}}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2504069538682520235663395798110}{160041699537014921582740396107}}}{z}\right)\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107}\right)}{z}}} \]

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

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107}\right)}{z}}} \]

      7. metadata-eval 99.9

         \[\leadsto x + \frac{y}{14.431876219268936 + \frac{\color{blue}{-15.646356830292042}}{z}} \]

   7. Simplified 99.9%

      \[\leadsto x + \frac{y}{\color{blue}{14.431876219268936 + \frac{-15.646356830292042}{z}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right), \frac{y}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{14.431876219268936 + \frac{-15.646356830292042}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{14.431876219268936 + \frac{-15.646356830292042}{z}}\\ \mathbf{if}\;z \leq -4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.07852944389170011, -0.10095235035524991\right), 0.39999999996247915\right), 12.000000000000014\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (+ 14.431876219268936 (/ -15.646356830292042 z))))))
   (if (<= z -4.0)
     t_0
     (if (<= z 4.4e-20)
       (+
        x
        (/
         y
         (fma
          z
          (fma
           z
           (fma z 0.07852944389170011 -0.10095235035524991)
           0.39999999996247915)
          12.000000000000014)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / (14.431876219268936 + (-15.646356830292042 / z)));
	double tmp;
	if (z <= -4.0) {
		tmp = t_0;
	} else if (z <= 4.4e-20) {
		tmp = x + (y / fma(z, fma(z, fma(z, 0.07852944389170011, -0.10095235035524991), 0.39999999996247915), 12.000000000000014));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(14.431876219268936 + Float64(-15.646356830292042 / z))))
	tmp = 0.0
	if (z <= -4.0)
		tmp = t_0;
	elseif (z <= 4.4e-20)
		tmp = Float64(x + Float64(y / fma(z, fma(z, fma(z, 0.07852944389170011, -0.10095235035524991), 0.39999999996247915), 12.000000000000014)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(14.431876219268936 + N[(-15.646356830292042 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.0], t$95$0, If[LessEqual[z, 4.4e-20], N[(x + N[(y / N[(z * N[(z * N[(z * 0.07852944389170011 + -0.10095235035524991), $MachinePrecision] + 0.39999999996247915), $MachinePrecision] + 12.000000000000014), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -4 or 4.39999999999999982e-20 < z

   1. Initial program 43.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-lowering-+.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 59.1

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

   4. Applied egg-rr 59.1%

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

   5. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\color{blue}{\frac{10000000000000000}{692910599291889} - \frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot \frac{1}{z}}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot \frac{1}{z}\right)\right)}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot \frac{1}{z}\right)\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot 1}{z}}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2504069538682520235663395798110}{160041699537014921582740396107}}}{z}\right)\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107}\right)}{z}}} \]

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

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107}\right)}{z}}} \]

      7. metadata-eval 98.2

         \[\leadsto x + \frac{y}{14.431876219268936 + \frac{\color{blue}{-15.646356830292042}}{z}} \]

   7. Simplified 98.2%

      \[\leadsto x + \frac{y}{\color{blue}{14.431876219268936 + \frac{-15.646356830292042}{z}}} \]

   if -4 < z < 4.39999999999999982e-20

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-lowering-+.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 99.3

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\frac{3350343815022304}{279195317918525} + z \cdot \left(\frac{155900051080628738716045985239}{389750127738131234692690878125} + z \cdot \left(\frac{119290279017840661005388637518195067609906756376454090785993}{1519051620718896144731238710708768455317203651375836035156250} \cdot z - \frac{54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125}\right)\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{z \cdot \left(\frac{155900051080628738716045985239}{389750127738131234692690878125} + z \cdot \left(\frac{119290279017840661005388637518195067609906756376454090785993}{1519051620718896144731238710708768455317203651375836035156250} \cdot z - \frac{54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125}\right)\right) + \frac{3350343815022304}{279195317918525}}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \frac{155900051080628738716045985239}{389750127738131234692690878125} + z \cdot \left(\frac{119290279017840661005388637518195067609906756376454090785993}{1519051620718896144731238710708768455317203651375836035156250} \cdot z - \frac{54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125}\right), \frac{3350343815022304}{279195317918525}\right)}} \]

      3. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{119290279017840661005388637518195067609906756376454090785993}{1519051620718896144731238710708768455317203651375836035156250} \cdot z - \frac{54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125}\right) + \frac{155900051080628738716045985239}{389750127738131234692690878125}}, \frac{3350343815022304}{279195317918525}\right)} \]

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

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{119290279017840661005388637518195067609906756376454090785993}{1519051620718896144731238710708768455317203651375836035156250} \cdot z - \frac{54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125}, \frac{155900051080628738716045985239}{389750127738131234692690878125}\right)}, \frac{3350343815022304}{279195317918525}\right)} \]

      5. sub-neg N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{119290279017840661005388637518195067609906756376454090785993}{1519051620718896144731238710708768455317203651375836035156250} \cdot z + \left(\mathsf{neg}\left(\frac{54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125}\right)\right)}, \frac{155900051080628738716045985239}{389750127738131234692690878125}\right), \frac{3350343815022304}{279195317918525}\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{z \cdot \frac{119290279017840661005388637518195067609906756376454090785993}{1519051620718896144731238710708768455317203651375836035156250}} + \left(\mathsf{neg}\left(\frac{54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125}\right)\right), \frac{155900051080628738716045985239}{389750127738131234692690878125}\right), \frac{3350343815022304}{279195317918525}\right)} \]

      7. metadata-eval N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, z \cdot \frac{119290279017840661005388637518195067609906756376454090785993}{1519051620718896144731238710708768455317203651375836035156250} + \color{blue}{\frac{-54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125}}, \frac{155900051080628738716045985239}{389750127738131234692690878125}\right), \frac{3350343815022304}{279195317918525}\right)} \]

      8. accelerator-lowering-fma.f64 99.2

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, 0.07852944389170011, -0.10095235035524991\right)}, 0.39999999996247915\right), 12.000000000000014\right)} \]

   7. Simplified 99.2%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.07852944389170011, -0.10095235035524991\right), 0.39999999996247915\right), 12.000000000000014\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{14.431876219268936 + \frac{-15.646356830292042}{z}}\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.10095235035524991, 0.39999999996247915\right), 12.000000000000014\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (+ 14.431876219268936 (/ -15.646356830292042 z))))))
   (if (<= z -5.5)
     t_0
     (if (<= z 4.4e-20)
       (+
        x
        (/
         y
         (fma
          z
          (fma z -0.10095235035524991 0.39999999996247915)
          12.000000000000014)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / (14.431876219268936 + (-15.646356830292042 / z)));
	double tmp;
	if (z <= -5.5) {
		tmp = t_0;
	} else if (z <= 4.4e-20) {
		tmp = x + (y / fma(z, fma(z, -0.10095235035524991, 0.39999999996247915), 12.000000000000014));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(14.431876219268936 + Float64(-15.646356830292042 / z))))
	tmp = 0.0
	if (z <= -5.5)
		tmp = t_0;
	elseif (z <= 4.4e-20)
		tmp = Float64(x + Float64(y / fma(z, fma(z, -0.10095235035524991, 0.39999999996247915), 12.000000000000014)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(14.431876219268936 + N[(-15.646356830292042 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5], t$95$0, If[LessEqual[z, 4.4e-20], N[(x + N[(y / N[(z * N[(z * -0.10095235035524991 + 0.39999999996247915), $MachinePrecision] + 12.000000000000014), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 4.39999999999999982e-20 < z

   1. Initial program 43.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-lowering-+.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 59.1

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

   4. Applied egg-rr 59.1%

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

   5. Taylor expanded in z around inf

      \[\leadsto x + \frac{y}{\color{blue}{\frac{10000000000000000}{692910599291889} - \frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot \frac{1}{z}}} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot \frac{1}{z}\right)\right)}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot \frac{1}{z}\right)\right)}} \]

      3. associate-*r/ N/A

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\color{blue}{\frac{\frac{2504069538682520235663395798110}{160041699537014921582740396107} \cdot 1}{z}}\right)\right)} \]

      4. metadata-eval N/A

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{2504069538682520235663395798110}{160041699537014921582740396107}}}{z}\right)\right)} \]

      5. distribute-neg-frac N/A

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107}\right)}{z}}} \]

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

         \[\leadsto x + \frac{y}{\frac{10000000000000000}{692910599291889} + \color{blue}{\frac{\mathsf{neg}\left(\frac{2504069538682520235663395798110}{160041699537014921582740396107}\right)}{z}}} \]

      7. metadata-eval 98.2

         \[\leadsto x + \frac{y}{14.431876219268936 + \frac{\color{blue}{-15.646356830292042}}{z}} \]

   7. Simplified 98.2%

      \[\leadsto x + \frac{y}{\color{blue}{14.431876219268936 + \frac{-15.646356830292042}{z}}} \]

   if -5.5 < z < 4.39999999999999982e-20

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-lowering-+.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 99.3

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\frac{3350343815022304}{279195317918525} + z \cdot \left(\frac{155900051080628738716045985239}{389750127738131234692690878125} + \frac{-54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125} \cdot z\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{z \cdot \left(\frac{155900051080628738716045985239}{389750127738131234692690878125} + \frac{-54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125} \cdot z\right) + \frac{3350343815022304}{279195317918525}}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \frac{155900051080628738716045985239}{389750127738131234692690878125} + \frac{-54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125} \cdot z, \frac{3350343815022304}{279195317918525}\right)}} \]

      3. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{\frac{-54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125} \cdot z + \frac{155900051080628738716045985239}{389750127738131234692690878125}}, \frac{3350343815022304}{279195317918525}\right)} \]

      4. *-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125}} + \frac{155900051080628738716045985239}{389750127738131234692690878125}, \frac{3350343815022304}{279195317918525}\right)} \]

      5. accelerator-lowering-fma.f64 99.2

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.10095235035524991, 0.39999999996247915\right)}, 12.000000000000014\right)} \]

   7. Simplified 99.2%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.10095235035524991, 0.39999999996247915\right), 12.000000000000014\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-20}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.10095235035524991, 0.39999999996247915\right), 12.000000000000014\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)))
   (if (<= z -5.5)
     t_0
     (if (<= z 4.4e-20)
       (+
        x
        (/
         y
         (fma
          z
          (fma z -0.10095235035524991 0.39999999996247915)
          12.000000000000014)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	double tmp;
	if (z <= -5.5) {
		tmp = t_0;
	} else if (z <= 4.4e-20) {
		tmp = x + (y / fma(z, fma(z, -0.10095235035524991, 0.39999999996247915), 12.000000000000014));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x)
	tmp = 0.0
	if (z <= -5.5)
		tmp = t_0;
	elseif (z <= 4.4e-20)
		tmp = Float64(x + Float64(y / fma(z, fma(z, -0.10095235035524991, 0.39999999996247915), 12.000000000000014)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -5.5], t$95$0, If[LessEqual[z, 4.4e-20], N[(x + N[(y / N[(z * N[(z * -0.10095235035524991 + 0.39999999996247915), $MachinePrecision] + 12.000000000000014), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 4.39999999999999982e-20 < z

   1. Initial program 43.9%

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

   3. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

   5. Simplified 97.9%

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

   if -5.5 < z < 4.39999999999999982e-20

   1. Initial program 99.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-lowering-+.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 99.3

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in z around 0

      \[\leadsto x + \frac{y}{\color{blue}{\frac{3350343815022304}{279195317918525} + z \cdot \left(\frac{155900051080628738716045985239}{389750127738131234692690878125} + \frac{-54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125} \cdot z\right)}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{z \cdot \left(\frac{155900051080628738716045985239}{389750127738131234692690878125} + \frac{-54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125} \cdot z\right) + \frac{3350343815022304}{279195317918525}}} \]

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

         \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \frac{155900051080628738716045985239}{389750127738131234692690878125} + \frac{-54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125} \cdot z, \frac{3350343815022304}{279195317918525}\right)}} \]

      3. +-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{\frac{-54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125} \cdot z + \frac{155900051080628738716045985239}{389750127738131234692690878125}}, \frac{3350343815022304}{279195317918525}\right)} \]

      4. *-commutative N/A

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{z \cdot \frac{-54926362148836417054593262229306108478309176}{544082054113166395691471098236108147273828125}} + \frac{155900051080628738716045985239}{389750127738131234692690878125}, \frac{3350343815022304}{279195317918525}\right)} \]

      5. accelerator-lowering-fma.f64 99.2

         \[\leadsto x + \frac{y}{\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, -0.10095235035524991, 0.39999999996247915\right)}, 12.000000000000014\right)} \]

   7. Simplified 99.2%

      \[\leadsto x + \frac{y}{\color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.10095235035524991, 0.39999999996247915\right), 12.000000000000014\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.00277777777751721, 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)))
   (if (<= z -5.5)
     t_0
     (if (<= z 4.4e-20)
       (fma y (fma z -0.00277777777751721 0.08333333333333323) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	double tmp;
	if (z <= -5.5) {
		tmp = t_0;
	} else if (z <= 4.4e-20) {
		tmp = fma(y, fma(z, -0.00277777777751721, 0.08333333333333323), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x)
	tmp = 0.0
	if (z <= -5.5)
		tmp = t_0;
	elseif (z <= 4.4e-20)
		tmp = fma(y, fma(z, -0.00277777777751721, 0.08333333333333323), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -5.5], t$95$0, If[LessEqual[z, 4.4e-20], N[(y * N[(z * -0.00277777777751721 + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 4.39999999999999982e-20 < z

   1. Initial program 43.9%

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

   3. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

   5. Simplified 97.9%

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

   if -5.5 < z < 4.39999999999999982e-20

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 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. accelerator-lowering-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. *-commutative N/A

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

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

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

      11. metadata-eval 99.0

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

   5. Simplified 99.0%

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

Alternative 9: 98.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{14.431876219268936}\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.00277777777751721, 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y 14.431876219268936))))
   (if (<= z -5.5)
     t_0
     (if (<= z 4.4e-20)
       (fma y (fma z -0.00277777777751721 0.08333333333333323) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / 14.431876219268936);
	double tmp;
	if (z <= -5.5) {
		tmp = t_0;
	} else if (z <= 4.4e-20) {
		tmp = fma(y, fma(z, -0.00277777777751721, 0.08333333333333323), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / 14.431876219268936))
	tmp = 0.0
	if (z <= -5.5)
		tmp = t_0;
	elseif (z <= 4.4e-20)
		tmp = fma(y, fma(z, -0.00277777777751721, 0.08333333333333323), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / 14.431876219268936), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5], t$95$0, If[LessEqual[z, 4.4e-20], N[(y * N[(z * -0.00277777777751721 + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 4.39999999999999982e-20 < z

   1. Initial program 43.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

      8. +-lowering-+.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 59.1

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

   4. Applied egg-rr 59.1%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 97.5%

      \[\leadsto x + \frac{y}{\color{blue}{14.431876219268936}} \]

   if -5.5 < z < 4.39999999999999982e-20

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 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. accelerator-lowering-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. *-commutative N/A

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

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

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

      11. metadata-eval 99.0

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

   5. Simplified 99.0%

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

Alternative 10: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.00277777777751721, 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5)
   (fma y 0.0692910599291889 x)
   (if (<= z 4.4e-20)
     (fma y (fma z -0.00277777777751721 0.08333333333333323) x)
     (fma y 0.0692910599291889 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5) {
		tmp = fma(y, 0.0692910599291889, x);
	} else if (z <= 4.4e-20) {
		tmp = fma(y, fma(z, -0.00277777777751721, 0.08333333333333323), x);
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5)
		tmp = fma(y, 0.0692910599291889, x);
	elseif (z <= 4.4e-20)
		tmp = fma(y, fma(z, -0.00277777777751721, 0.08333333333333323), x);
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.5], N[(y * 0.0692910599291889 + x), $MachinePrecision], If[LessEqual[z, 4.4e-20], N[(y * N[(z * -0.00277777777751721 + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 4.39999999999999982e-20 < z

   1. Initial program 43.9%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 97.2

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

   5. Simplified 97.2%

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

   if -5.5 < z < 4.39999999999999982e-20

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 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. accelerator-lowering-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. *-commutative N/A

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

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

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

      11. metadata-eval 99.0

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

   5. Simplified 99.0%

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

Alternative 11: 98.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-20}:\\ \;\;\;\;x + y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5)
   (fma y 0.0692910599291889 x)
   (if (<= z 4.4e-20)
     (+ x (* y 0.08333333333333323))
     (fma y 0.0692910599291889 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5) {
		tmp = fma(y, 0.0692910599291889, x);
	} else if (z <= 4.4e-20) {
		tmp = x + (y * 0.08333333333333323);
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5)
		tmp = fma(y, 0.0692910599291889, x);
	elseif (z <= 4.4e-20)
		tmp = Float64(x + Float64(y * 0.08333333333333323));
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.5], N[(y * 0.0692910599291889 + x), $MachinePrecision], If[LessEqual[z, 4.4e-20], N[(x + N[(y * 0.08333333333333323), $MachinePrecision]), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 4.39999999999999982e-20 < z

   1. Initial program 43.9%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 97.2

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

   5. Simplified 97.2%

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

   if -5.5 < z < 4.39999999999999982e-20

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 98.8

         \[\leadsto x + \color{blue}{y \cdot 0.08333333333333323} \]

   5. Simplified 98.8%

      \[\leadsto x + \color{blue}{y \cdot 0.08333333333333323} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 98.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.08333333333333323, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5)
   (fma y 0.0692910599291889 x)
   (if (<= z 4.4e-20)
     (fma y 0.08333333333333323 x)
     (fma y 0.0692910599291889 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5) {
		tmp = fma(y, 0.0692910599291889, x);
	} else if (z <= 4.4e-20) {
		tmp = fma(y, 0.08333333333333323, x);
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5)
		tmp = fma(y, 0.0692910599291889, x);
	elseif (z <= 4.4e-20)
		tmp = fma(y, 0.08333333333333323, x);
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.5], N[(y * 0.0692910599291889 + x), $MachinePrecision], If[LessEqual[z, 4.4e-20], N[(y * 0.08333333333333323 + x), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 4.39999999999999982e-20 < z

   1. Initial program 43.9%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 97.2

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

   5. Simplified 97.2%

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

   if -5.5 < z < 4.39999999999999982e-20

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 98.8

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

   5. Simplified 98.8%

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

Alternative 13: 56.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-174}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.1e+95) x (if (<= x 7.5e-174) (* y 0.08333333333333323) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e+95) {
		tmp = x;
	} else if (x <= 7.5e-174) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = x;
	}
	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 <= (-2.1d+95)) then
        tmp = x
    else if (x <= 7.5d-174) then
        tmp = y * 0.08333333333333323d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.1e+95) {
		tmp = x;
	} else if (x <= 7.5e-174) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.1e+95:
		tmp = x
	elif x <= 7.5e-174:
		tmp = y * 0.08333333333333323
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.1e+95)
		tmp = x;
	elseif (x <= 7.5e-174)
		tmp = Float64(y * 0.08333333333333323);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.1e+95)
		tmp = x;
	elseif (x <= 7.5e-174)
		tmp = y * 0.08333333333333323;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.1e+95], x, If[LessEqual[x, 7.5e-174], N[(y * 0.08333333333333323), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e95 or 7.5000000000000003e-174 < x

   1. Initial program 70.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 70.8%

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

   if -2.1e95 < x < 7.5000000000000003e-174

   1. Initial program 75.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 73.0

         \[\leadsto x + \color{blue}{y \cdot 0.08333333333333323} \]

   5. Simplified 73.0%

      \[\leadsto x + \color{blue}{y \cdot 0.08333333333333323} \]

   6. Taylor expanded in x around 0

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

      1. *-lowering-*.f64 52.4

         \[\leadsto \color{blue}{0.08333333333333323 \cdot y} \]

   8. Simplified 52.4%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-174}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.4% accurate, 47.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

function code(x, y, z)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 72.8%

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

3. Taylor expanded in x around inf

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

5. Simplified 48.5%

   \[\leadsto \color{blue}{x} \]
6. 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 2024195 
(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))))