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

Percentage Accurate: 69.4% → 99.5%

Time: 9.4s

Alternatives: 13

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        y
        (+
         (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
         0.279195317918525))
       (+ (* (+ z 6.012459259764103) z) 3.350343815022304))
      5e+295)
   (fma
    (/
     (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
     (fma (+ 6.012459259764103 z) z 3.350343815022304))
    y
    x)
   (+ x (/ y 14.431876219268936))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304)) <= 5e+295) {
		tmp = fma((fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma((6.012459259764103 + z), z, 3.350343815022304)), y, x);
	} else {
		tmp = x + (y / 14.431876219268936);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304)) <= 5e+295)
		tmp = fma(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma(Float64(6.012459259764103 + z), z, 3.350343815022304)), y, x);
	else
		tmp = Float64(x + Float64(y / 14.431876219268936));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[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], 5e+295], N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] / N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(y / 14.431876219268936), $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))) < 4.99999999999999991e295

   1. Initial program 93.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   if 4.99999999999999991e295 < (/.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. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 18.3

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 18.3

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 18.3

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 18.3

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 18.3

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

   4. Applied rewrites 18.3%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 100.0%

      \[\leadsto x + \frac{y}{\color{blue}{14.431876219268936}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+55} \lor \neg \left(t\_0 \leq 5 \cdot 10^{+295}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
   (if (or (<= t_0 5e+55) (not (<= t_0 5e+295)))
     (fma 0.0692910599291889 y x)
     (* 0.08333333333333323 y))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if ((t_0 <= 5e+55) || !(t_0 <= 5e+295)) {
		tmp = fma(0.0692910599291889, y, x);
	} else {
		tmp = 0.08333333333333323 * y;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))
	tmp = 0.0
	if ((t_0 <= 5e+55) || !(t_0 <= 5e+295))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = Float64(0.08333333333333323 * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 5e+55], N[Not[LessEqual[t$95$0, 5e+295]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(0.08333333333333323 * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 5.00000000000000046e55 or 4.99999999999999991e295 < (/.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 66.5%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if 5.00000000000000046e55 < (/.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))) < 4.99999999999999991e295

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 94.1

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

   7. Applied rewrites 94.1%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 83.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 5 \cdot 10^{+55} \lor \neg \left(\frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 5 \cdot 10^{+295}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5 \lor \neg \left(z \leq 3.3\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.07852944389170011, z, -0.10095235035524991\right), z, 0.39999999996247915\right), z, 12.000000000000014\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -6.5) (not (<= z 3.3)))
   (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)
   (+
    x
    (/
     y
     (fma
      (fma
       (fma 0.07852944389170011 z -0.10095235035524991)
       z
       0.39999999996247915)
      z
      12.000000000000014)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -6.5) || !(z <= 3.3)) {
		tmp = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	} else {
		tmp = x + (y / fma(fma(fma(0.07852944389170011, z, -0.10095235035524991), z, 0.39999999996247915), z, 12.000000000000014));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -6.5) || !(z <= 3.3))
		tmp = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x);
	else
		tmp = Float64(x + Float64(y / fma(fma(fma(0.07852944389170011, z, -0.10095235035524991), z, 0.39999999996247915), z, 12.000000000000014)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -6.5], N[Not[LessEqual[z, 3.3]], $MachinePrecision]], N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(y / N[(N[(N[(0.07852944389170011 * z + -0.10095235035524991), $MachinePrecision] * z + 0.39999999996247915), $MachinePrecision] * z + 12.000000000000014), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5 or 3.2999999999999998 < z

   1. Initial program 38.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

   5. Applied rewrites 99.1%

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

   if -6.5 < z < 3.2999999999999998

   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. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.3

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 99.3

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.3

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 99.3

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \lor \neg \left(z \leq 3.3\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.07852944389170011, z, -0.10095235035524991\right), z, 0.39999999996247915\right), z, 12.000000000000014\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.5:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889 - \frac{\frac{0.4046220386999212}{z} - 0.07512208616047561}{z}, y, x\right)\\ \mathbf{elif}\;z \leq 3.3:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.07852944389170011, z, -0.10095235035524991\right), z, 0.39999999996247915\right), z, 12.000000000000014\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -6.5)
   (fma
    (-
     0.0692910599291889
     (/ (- (/ 0.4046220386999212 z) 0.07512208616047561) z))
    y
    x)
   (if (<= z 3.3)
     (+
      x
      (/
       y
       (fma
        (fma
         (fma 0.07852944389170011 z -0.10095235035524991)
         z
         0.39999999996247915)
        z
        12.000000000000014)))
     (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -6.5) {
		tmp = fma((0.0692910599291889 - (((0.4046220386999212 / z) - 0.07512208616047561) / z)), y, x);
	} else if (z <= 3.3) {
		tmp = x + (y / fma(fma(fma(0.07852944389170011, z, -0.10095235035524991), z, 0.39999999996247915), z, 12.000000000000014));
	} else {
		tmp = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -6.5)
		tmp = fma(Float64(0.0692910599291889 - Float64(Float64(Float64(0.4046220386999212 / z) - 0.07512208616047561) / z)), y, x);
	elseif (z <= 3.3)
		tmp = Float64(x + Float64(y / fma(fma(fma(0.07852944389170011, z, -0.10095235035524991), z, 0.39999999996247915), z, 12.000000000000014)));
	else
		tmp = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -6.5], N[(N[(0.0692910599291889 - N[(N[(N[(0.4046220386999212 / z), $MachinePrecision] - 0.07512208616047561), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 3.3], N[(x + N[(y / N[(N[(N[(0.07852944389170011 * z + -0.10095235035524991), $MachinePrecision] * z + 0.39999999996247915), $MachinePrecision] * z + 12.000000000000014), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5

   1. Initial program 45.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 58.2%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{692910599291889}{10000000000000000} + \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}\right) - \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}}{{z}^{2}}}, y, x\right) \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{692910599291889}{10000000000000000} + \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}\right) + \left(\mathsf{neg}\left(\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}}{{z}^{2}}\right)\right)}, y, x\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}}{{z}^{2}}\right)\right) + \left(\frac{692910599291889}{10000000000000000} + \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}\right)}, y, x\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}}{{z}^{2}}\right)\right) + \color{blue}{\left(\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z} + \frac{692910599291889}{10000000000000000}\right)}, y, x\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}}{{z}^{2}}\right)\right) + \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}\right) + \frac{692910599291889}{10000000000000000}}, y, x\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(0 - \frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}}{{z}^{2}}\right)} + \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}\right) + \frac{692910599291889}{10000000000000000}, y, x\right) \]

      6. associate--r- N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - \left(\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}}{{z}^{2}} - \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}\right)\right)} + \frac{692910599291889}{10000000000000000}, y, x\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(0 - \left(\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}}{\color{blue}{z \cdot z}} - \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}\right)\right) + \frac{692910599291889}{10000000000000000}, y, x\right) \]

      8. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\left(0 - \left(\color{blue}{\frac{\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000}}{z}}{z}} - \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}\right)\right) + \frac{692910599291889}{10000000000000000}, y, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(0 - \left(\frac{\frac{\color{blue}{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot 1}}{z}}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}\right)\right) + \frac{692910599291889}{10000000000000000}, y, x\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\left(0 - \left(\frac{\color{blue}{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z}}}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot \frac{1}{z}\right)\right) + \frac{692910599291889}{10000000000000000}, y, x\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\left(0 - \left(\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z}}{z} - \color{blue}{\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000} \cdot 1}{z}}\right)\right) + \frac{692910599291889}{10000000000000000}, y, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\left(0 - \left(\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z}}{z} - \frac{\color{blue}{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}}{z}\right)\right) + \frac{692910599291889}{10000000000000000}, y, x\right) \]

      13. div-sub N/A

          \[\leadsto \mathsf{fma}\left(\left(0 - \color{blue}{\frac{\frac{4046220386999211718548694042263781576003973599}{10000000000000000000000000000000000000000000000} \cdot \frac{1}{z} - \frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z}}\right) + \frac{692910599291889}{10000000000000000}, y, x\right) \]

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

   7. Applied rewrites 99.0%

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

   if -6.5 < z < 3.2999999999999998

   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. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.3

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 99.3

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 99.3

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 99.3

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if 3.2999999999999998 < z

   1. Initial program 30.9%

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

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 3\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005951669793454025, z, 0.0007936505811533442\right), z, -0.00277777777751721\right), z, 0.08333333333333323\right), y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5) (not (<= z 3.0)))
   (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)
   (fma
    (fma
     (fma
      (fma -0.0005951669793454025 z 0.0007936505811533442)
      z
      -0.00277777777751721)
     z
     0.08333333333333323)
    y
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5) || !(z <= 3.0)) {
		tmp = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	} else {
		tmp = fma(fma(fma(fma(-0.0005951669793454025, z, 0.0007936505811533442), z, -0.00277777777751721), z, 0.08333333333333323), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.5) || !(z <= 3.0))
		tmp = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x);
	else
		tmp = fma(fma(fma(fma(-0.0005951669793454025, z, 0.0007936505811533442), z, -0.00277777777751721), z, 0.08333333333333323), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.5], N[Not[LessEqual[z, 3.0]], $MachinePrecision]], N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(N[(-0.0005951669793454025 * z + 0.0007936505811533442), $MachinePrecision] * z + -0.00277777777751721), $MachinePrecision] * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 3 < z

   1. Initial program 38.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

   5. Applied rewrites 99.1%

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

   if -5.5 < z < 3

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 99.4

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

   7. Applied rewrites 99.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 3\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005951669793454025, z, 0.0007936505811533442\right), z, -0.00277777777751721\right), z, 0.08333333333333323\right), y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 4.4\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0007936505811533442, z, -0.00277777777751721\right), z, 0.08333333333333323\right), y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5) (not (<= z 4.4)))
   (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)
   (fma
    (fma
     (fma 0.0007936505811533442 z -0.00277777777751721)
     z
     0.08333333333333323)
    y
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5) || !(z <= 4.4)) {
		tmp = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	} else {
		tmp = fma(fma(fma(0.0007936505811533442, z, -0.00277777777751721), z, 0.08333333333333323), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.5) || !(z <= 4.4))
		tmp = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x);
	else
		tmp = fma(fma(fma(0.0007936505811533442, z, -0.00277777777751721), z, 0.08333333333333323), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.5], N[Not[LessEqual[z, 4.4]], $MachinePrecision]], N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(0.0007936505811533442 * z + -0.00277777777751721), $MachinePrecision] * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 4.4000000000000004 < z

   1. Initial program 38.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. times-frac N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. mul-1-neg N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. +-commutative N/A

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

   5. Applied rewrites 99.1%

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

   if -5.5 < z < 4.4000000000000004

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 99.4

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

   7. Applied rewrites 99.4%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 4.4\right):\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0007936505811533442, z, -0.00277777777751721\right), z, 0.08333333333333323\right), y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 4.2\right):\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0007936505811533442, z, -0.00277777777751721\right), z, 0.08333333333333323\right), y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5) (not (<= z 4.2)))
   (+ x (/ y 14.431876219268936))
   (fma
    (fma
     (fma 0.0007936505811533442 z -0.00277777777751721)
     z
     0.08333333333333323)
    y
    x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5) || !(z <= 4.2)) {
		tmp = x + (y / 14.431876219268936);
	} else {
		tmp = fma(fma(fma(0.0007936505811533442, z, -0.00277777777751721), z, 0.08333333333333323), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.5) || !(z <= 4.2))
		tmp = Float64(x + Float64(y / 14.431876219268936));
	else
		tmp = fma(fma(fma(0.0007936505811533442, z, -0.00277777777751721), z, 0.08333333333333323), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.5], N[Not[LessEqual[z, 4.2]], $MachinePrecision]], N[(x + N[(y / 14.431876219268936), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0007936505811533442 * z + -0.00277777777751721), $MachinePrecision] * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 4.20000000000000018 < z

   1. Initial program 38.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 56.7

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 56.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 56.7

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 56.7

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 56.7

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

   4. Applied rewrites 56.7%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 98.7%

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

   if -5.5 < z < 4.20000000000000018

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 99.4

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

   7. Applied rewrites 99.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 4.2\right):\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0007936505811533442, z, -0.00277777777751721\right), z, 0.08333333333333323\right), y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 5.2\right):\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5) (not (<= z 5.2)))
   (+ x (/ y 14.431876219268936))
   (fma y (fma -0.00277777777751721 z 0.08333333333333323) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5) || !(z <= 5.2)) {
		tmp = x + (y / 14.431876219268936);
	} else {
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.5) || !(z <= 5.2))
		tmp = Float64(x + Float64(y / 14.431876219268936));
	else
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.5], N[Not[LessEqual[z, 5.2]], $MachinePrecision]], N[(x + N[(y / 14.431876219268936), $MachinePrecision]), $MachinePrecision], N[(y * N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 5.20000000000000018 < z

   1. Initial program 38.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 56.7

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 56.7

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 56.7

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 56.7

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

      17. lift-+.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 56.7

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

   4. Applied rewrites 56.7%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 98.7%

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

   if -5.5 < z < 5.20000000000000018

   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. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 5.2\right):\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 5.2\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5) (not (<= z 5.2)))
   (fma 0.0692910599291889 y x)
   (fma y (fma -0.00277777777751721 z 0.08333333333333323) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5.5) || !(z <= 5.2)) {
		tmp = fma(0.0692910599291889, y, x);
	} else {
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.5) || !(z <= 5.2))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = fma(y, fma(-0.00277777777751721, z, 0.08333333333333323), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.5], N[Not[LessEqual[z, 5.2]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(y * N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 5.20000000000000018 < z

   1. Initial program 38.5%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if -5.5 < z < 5.20000000000000018

   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. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 5.2\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+81}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+55}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+194}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.9e+81)
   (* 0.0692910599291889 y)
   (if (<= y 2.05e+55)
     (* 1.0 x)
     (if (<= y 1.25e+194)
       (* 0.08333333333333323 y)
       (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.9e+81) {
		tmp = 0.0692910599291889 * y;
	} else if (y <= 2.05e+55) {
		tmp = 1.0 * x;
	} else if (y <= 1.25e+194) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-2.9d+81)) then
        tmp = 0.0692910599291889d0 * y
    else if (y <= 2.05d+55) then
        tmp = 1.0d0 * x
    else if (y <= 1.25d+194) then
        tmp = 0.08333333333333323d0 * y
    else
        tmp = 0.0692910599291889d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.9e+81) {
		tmp = 0.0692910599291889 * y;
	} else if (y <= 2.05e+55) {
		tmp = 1.0 * x;
	} else if (y <= 1.25e+194) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.9e+81:
		tmp = 0.0692910599291889 * y
	elif y <= 2.05e+55:
		tmp = 1.0 * x
	elif y <= 1.25e+194:
		tmp = 0.08333333333333323 * y
	else:
		tmp = 0.0692910599291889 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.9e+81)
		tmp = Float64(0.0692910599291889 * y);
	elseif (y <= 2.05e+55)
		tmp = Float64(1.0 * x);
	elseif (y <= 1.25e+194)
		tmp = Float64(0.08333333333333323 * y);
	else
		tmp = Float64(0.0692910599291889 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.9e+81)
		tmp = 0.0692910599291889 * y;
	elseif (y <= 2.05e+55)
		tmp = 1.0 * x;
	elseif (y <= 1.25e+194)
		tmp = 0.08333333333333323 * y;
	else
		tmp = 0.0692910599291889 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.9e+81], N[(0.0692910599291889 * y), $MachinePrecision], If[LessEqual[y, 2.05e+55], N[(1.0 * x), $MachinePrecision], If[LessEqual[y, 1.25e+194], N[(0.08333333333333323 * y), $MachinePrecision], N[(0.0692910599291889 * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9e81 or 1.24999999999999997e194 < y

   1. Initial program 55.7%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.9%

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

   if -2.9e81 < y < 2.04999999999999991e55

   1. Initial program 76.4%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 86.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 69.4%

       \[\leadsto 1 \cdot x \]

   if 2.04999999999999991e55 < y < 1.24999999999999997e194

   1. Initial program 73.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 86.3%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-+.f64 77.8

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

   7. Applied rewrites 77.8%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 61.2%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+81}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+55}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+194}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 5.5\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5.5) (not (<= z 5.5)))
   (fma 0.0692910599291889 y x)
   (fma 0.08333333333333323 y x)))

C

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

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5.5) || !(z <= 5.5))
		tmp = fma(0.0692910599291889, y, x);
	else
		tmp = fma(0.08333333333333323, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5.5], N[Not[LessEqual[z, 5.5]], $MachinePrecision]], N[(0.0692910599291889 * y + x), $MachinePrecision], N[(0.08333333333333323 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 5.5 < z

   1. Initial program 38.5%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.4

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

   5. Applied rewrites 98.4%

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

   if -5.5 < z < 5.5

   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. lower-fma.f64 98.6

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

   5. Applied rewrites 98.6%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \lor \neg \left(z \leq 5.5\right):\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+81} \lor \neg \left(y \leq 6.2 \cdot 10^{+55}\right):\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.9e+81) (not (<= y 6.2e+55)))
   (* 0.0692910599291889 y)
   (* 1.0 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.9e+81) || !(y <= 6.2e+55)) {
		tmp = 0.0692910599291889 * y;
	} else {
		tmp = 1.0 * 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 ((y <= (-2.9d+81)) .or. (.not. (y <= 6.2d+55))) then
        tmp = 0.0692910599291889d0 * y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.9e+81) || !(y <= 6.2e+55)) {
		tmp = 0.0692910599291889 * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.9e+81) or not (y <= 6.2e+55):
		tmp = 0.0692910599291889 * y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.9e+81) || !(y <= 6.2e+55))
		tmp = Float64(0.0692910599291889 * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.9e+81) || ~((y <= 6.2e+55)))
		tmp = 0.0692910599291889 * y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.9e+81], N[Not[LessEqual[y, 6.2e+55]], $MachinePrecision]], N[(0.0692910599291889 * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9e81 or 6.19999999999999987e55 < y

   1. Initial program 60.9%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. lower-fma.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.8%

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

   if -2.9e81 < y < 6.19999999999999987e55

   1. Initial program 76.4%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 86.4%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 69.4%

       \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+81} \lor \neg \left(y \leq 6.2 \cdot 10^{+55}\right):\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 31.0% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.2%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. lower-fma.f64 79.0

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

5. Applied rewrites 79.0%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 33.4%

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

Developer Target 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024324 
(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))))