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

Percentage Accurate: 68.3% → 98.9%

Time: 12.9s

Alternatives: 11

Speedup: 2.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.3% 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: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        y
        (+
         (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
         0.279195317918525))
       (+ (* z (+ z 6.012459259764103)) 3.350343815022304))
      5e+279)
   (fma
    (/
     (fma z (fma z 0.0692910599291889 0.4917317610505968) 0.279195317918525)
     (fma z (+ z 6.012459259764103) 3.350343815022304))
    y
    x)
   (+ x (/ y 14.431876219268936))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304)) <= 5e+279) {
		tmp = fma((fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525) / fma(z, (z + 6.012459259764103), 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(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304)) <= 5e+279)
		tmp = fma(Float64(fma(z, fma(z, 0.0692910599291889, 0.4917317610505968), 0.279195317918525) / fma(z, Float64(z + 6.012459259764103), 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[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision], 5e+279], N[(N[(N[(z * N[(z * 0.0692910599291889 + 0.4917317610505968), $MachinePrecision] + 0.279195317918525), $MachinePrecision] / N[(z * N[(z + 6.012459259764103), $MachinePrecision] + 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))) < 5.0000000000000002e279

   1. Initial program 98.7%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   if 5.0000000000000002e279 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))

   1. Initial program 0.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 9.5

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

   4. Applied egg-rr 9.5%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq 5 \cdot 10^{+279}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(z, \mathsf{fma}\left(z, 0.0692910599291889, 0.4917317610505968\right), 0.279195317918525\right)}{\mathsf{fma}\left(z, z + 6.012459259764103, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{14.431876219268936}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq -2 \cdot 10^{+232}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (/
       (*
        y
        (+
         (* z (+ (* z 0.0692910599291889) 0.4917317610505968))
         0.279195317918525))
       (+ (* z (+ z 6.012459259764103)) 3.350343815022304))
      -2e+232)
   (* y 0.08333333333333323)
   (fma y 0.0692910599291889 x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((y * ((z * ((z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / ((z * (z + 6.012459259764103)) + 3.350343815022304)) <= -2e+232) {
		tmp = y * 0.08333333333333323;
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(y * Float64(Float64(z * Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968)) + 0.279195317918525)) / Float64(Float64(z * Float64(z + 6.012459259764103)) + 3.350343815022304)) <= -2e+232)
		tmp = Float64(y * 0.08333333333333323);
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(y * N[(N[(z * N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision]), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(z * N[(z + 6.012459259764103), $MachinePrecision]), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision], -2e+232], N[(y * 0.08333333333333323), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -2.00000000000000011e232

   1. Initial program 86.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 98.7

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in z around 0

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

   7. Simplified 83.1%

      \[\leadsto x + \frac{y}{\color{blue}{12.000000000000014}} \]

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 83.1

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

   10. Simplified 83.1%

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

   if -2.00000000000000011e232 < (/.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 69.7%

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

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

      3. accelerator-lowering-fma.f64 85.0

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

   5. Simplified 85.0%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z \cdot \left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) + 0.279195317918525\right)}{z \cdot \left(z + 6.012459259764103\right) + 3.350343815022304} \leq -2 \cdot 10^{+232}:\\ \;\;\;\;y \cdot 0.08333333333333323\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{14.431876219268936 + \frac{-15.646356830292042}{z}}\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.10095235035524991, 0.39999999996247915\right), 12.000000000000014\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y (+ 14.431876219268936 (/ -15.646356830292042 z))))))
   (if (<= z -5.5)
     t_0
     (if (<= z 1.3e-6)
       (+
        x
        (/
         y
         (fma
          z
          (fma z -0.10095235035524991 0.39999999996247915)
          12.000000000000014)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / (14.431876219268936 + (-15.646356830292042 / z)));
	double tmp;
	if (z <= -5.5) {
		tmp = t_0;
	} else if (z <= 1.3e-6) {
		tmp = x + (y / fma(z, fma(z, -0.10095235035524991, 0.39999999996247915), 12.000000000000014));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / Float64(14.431876219268936 + Float64(-15.646356830292042 / z))))
	tmp = 0.0
	if (z <= -5.5)
		tmp = t_0;
	elseif (z <= 1.3e-6)
		tmp = Float64(x + Float64(y / fma(z, fma(z, -0.10095235035524991, 0.39999999996247915), 12.000000000000014)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / N[(14.431876219268936 + N[(-15.646356830292042 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5], t$95$0, If[LessEqual[z, 1.3e-6], N[(x + N[(y / N[(z * N[(z * -0.10095235035524991 + 0.39999999996247915), $MachinePrecision] + 12.000000000000014), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 1.30000000000000005e-6 < z

   1. Initial program 41.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 48.2

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

   4. Applied egg-rr 48.2%

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

   5. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

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

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. distribute-neg-frac N/A

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

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

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

      7. metadata-eval 99.8

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

   7. Simplified 99.8%

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

   if -5.5 < z < 1.30000000000000005e-6

   1. Initial program 99.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 99.4

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

   7. Simplified 99.4%

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

Alternative 4: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, -0.10095235035524991, 0.39999999996247915\right), 12.000000000000014\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)))
   (if (<= z -5.5)
     t_0
     (if (<= z 1.3e-6)
       (+
        x
        (/
         y
         (fma
          z
          (fma z -0.10095235035524991 0.39999999996247915)
          12.000000000000014)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	double tmp;
	if (z <= -5.5) {
		tmp = t_0;
	} else if (z <= 1.3e-6) {
		tmp = x + (y / fma(z, fma(z, -0.10095235035524991, 0.39999999996247915), 12.000000000000014));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x)
	tmp = 0.0
	if (z <= -5.5)
		tmp = t_0;
	elseif (z <= 1.3e-6)
		tmp = Float64(x + Float64(y / fma(z, fma(z, -0.10095235035524991, 0.39999999996247915), 12.000000000000014)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -5.5], t$95$0, If[LessEqual[z, 1.3e-6], N[(x + N[(y / N[(z * N[(z * -0.10095235035524991 + 0.39999999996247915), $MachinePrecision] + 12.000000000000014), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 1.30000000000000005e-6 < z

   1. Initial program 41.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. Simplified 99.5%

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

   if -5.5 < z < 1.30000000000000005e-6

   1. Initial program 99.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

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

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 99.4

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

   7. Simplified 99.4%

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

Alternative 5: 99.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, 0.0692910599291889 - \frac{-0.07512208616047561}{z}, x\right)\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, 0.39999999996247915, 12.000000000000014\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma y (- 0.0692910599291889 (/ -0.07512208616047561 z)) x)))
   (if (<= z -5.5)
     t_0
     (if (<= z 1.3e-6)
       (+ x (/ y (fma z 0.39999999996247915 12.000000000000014)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma(y, (0.0692910599291889 - (-0.07512208616047561 / z)), x);
	double tmp;
	if (z <= -5.5) {
		tmp = t_0;
	} else if (z <= 1.3e-6) {
		tmp = x + (y / fma(z, 0.39999999996247915, 12.000000000000014));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(y, Float64(0.0692910599291889 - Float64(-0.07512208616047561 / z)), x)
	tmp = 0.0
	if (z <= -5.5)
		tmp = t_0;
	elseif (z <= 1.3e-6)
		tmp = Float64(x + Float64(y / fma(z, 0.39999999996247915, 12.000000000000014)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(0.0692910599291889 - N[(-0.07512208616047561 / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -5.5], t$95$0, If[LessEqual[z, 1.3e-6], N[(x + N[(y / N[(z * 0.39999999996247915 + 12.000000000000014), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 1.30000000000000005e-6 < z

   1. Initial program 41.5%

      \[x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \]
   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. Simplified 99.5%

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

   if -5.5 < z < 1.30000000000000005e-6

   1. Initial program 99.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{155900051080628738716045985239}{389750127738131234692690878125} \cdot z + \frac{3350343815022304}{279195317918525}}} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{z \cdot \frac{155900051080628738716045985239}{389750127738131234692690878125}} + \frac{3350343815022304}{279195317918525}} \]

      3. accelerator-lowering-fma.f64 99.3

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

   7. Simplified 99.3%

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

Alternative 6: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{14.431876219268936}\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;x + \frac{y}{\mathsf{fma}\left(z, 0.39999999996247915, 12.000000000000014\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y 14.431876219268936))))
   (if (<= z -5.5)
     t_0
     (if (<= z 1.3e-6)
       (+ x (/ y (fma z 0.39999999996247915 12.000000000000014)))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / 14.431876219268936);
	double tmp;
	if (z <= -5.5) {
		tmp = t_0;
	} else if (z <= 1.3e-6) {
		tmp = x + (y / fma(z, 0.39999999996247915, 12.000000000000014));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / 14.431876219268936))
	tmp = 0.0
	if (z <= -5.5)
		tmp = t_0;
	elseif (z <= 1.3e-6)
		tmp = Float64(x + Float64(y / fma(z, 0.39999999996247915, 12.000000000000014)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / 14.431876219268936), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5], t$95$0, If[LessEqual[z, 1.3e-6], N[(x + N[(y / N[(z * 0.39999999996247915 + 12.000000000000014), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 1.30000000000000005e-6 < z

   1. Initial program 41.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 48.2

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

   4. Applied egg-rr 48.2%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 99.2%

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

   if -5.5 < z < 1.30000000000000005e-6

   1. Initial program 99.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 99.4

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{\frac{155900051080628738716045985239}{389750127738131234692690878125} \cdot z + \frac{3350343815022304}{279195317918525}}} \]

      2. *-commutative N/A

         \[\leadsto x + \frac{y}{\color{blue}{z \cdot \frac{155900051080628738716045985239}{389750127738131234692690878125}} + \frac{3350343815022304}{279195317918525}} \]

      3. accelerator-lowering-fma.f64 99.3

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

   7. Simplified 99.3%

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

Alternative 7: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \frac{y}{14.431876219268936}\\ \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.00277777777751721, 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ x (/ y 14.431876219268936))))
   (if (<= z -5.5)
     t_0
     (if (<= z 1.3e-6)
       (fma y (fma z -0.00277777777751721 0.08333333333333323) x)
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x + (y / 14.431876219268936);
	double tmp;
	if (z <= -5.5) {
		tmp = t_0;
	} else if (z <= 1.3e-6) {
		tmp = fma(y, fma(z, -0.00277777777751721, 0.08333333333333323), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(x + Float64(y / 14.431876219268936))
	tmp = 0.0
	if (z <= -5.5)
		tmp = t_0;
	elseif (z <= 1.3e-6)
		tmp = fma(y, fma(z, -0.00277777777751721, 0.08333333333333323), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x + N[(y / 14.431876219268936), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5], t$95$0, If[LessEqual[z, 1.3e-6], N[(y * N[(z * -0.00277777777751721 + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 1.30000000000000005e-6 < z

   1. Initial program 41.5%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. accelerator-lowering-fma.f64 48.2

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

   4. Applied egg-rr 48.2%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 99.2%

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

   if -5.5 < z < 1.30000000000000005e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval 99.3

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

   5. Simplified 99.3%

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

Alternative 8: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(z, -0.00277777777751721, 0.08333333333333323\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5)
   (fma y 0.0692910599291889 x)
   (if (<= z 1.3e-6)
     (fma y (fma z -0.00277777777751721 0.08333333333333323) x)
     (fma y 0.0692910599291889 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5) {
		tmp = fma(y, 0.0692910599291889, x);
	} else if (z <= 1.3e-6) {
		tmp = fma(y, fma(z, -0.00277777777751721, 0.08333333333333323), x);
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5)
		tmp = fma(y, 0.0692910599291889, x);
	elseif (z <= 1.3e-6)
		tmp = fma(y, fma(z, -0.00277777777751721, 0.08333333333333323), x);
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.5], N[(y * 0.0692910599291889 + x), $MachinePrecision], If[LessEqual[z, 1.3e-6], N[(y * N[(z * -0.00277777777751721 + 0.08333333333333323), $MachinePrecision] + x), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 1.30000000000000005e-6 < z

   1. Initial program 41.5%

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

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

      3. accelerator-lowering-fma.f64 98.9

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

   5. Simplified 98.9%

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

   if -5.5 < z < 1.30000000000000005e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-out N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval 99.3

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

   5. Simplified 99.3%

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

Alternative 9: 98.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.08333333333333323, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 0.0692910599291889, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.5)
   (fma y 0.0692910599291889 x)
   (if (<= z 1.3e-6)
     (fma y 0.08333333333333323 x)
     (fma y 0.0692910599291889 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.5) {
		tmp = fma(y, 0.0692910599291889, x);
	} else if (z <= 1.3e-6) {
		tmp = fma(y, 0.08333333333333323, x);
	} else {
		tmp = fma(y, 0.0692910599291889, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.5)
		tmp = fma(y, 0.0692910599291889, x);
	elseif (z <= 1.3e-6)
		tmp = fma(y, 0.08333333333333323, x);
	else
		tmp = fma(y, 0.0692910599291889, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.5], N[(y * 0.0692910599291889 + x), $MachinePrecision], If[LessEqual[z, 1.3e-6], N[(y * 0.08333333333333323 + x), $MachinePrecision], N[(y * 0.0692910599291889 + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5 or 1.30000000000000005e-6 < z

   1. Initial program 41.5%

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

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

      3. accelerator-lowering-fma.f64 98.9

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

   5. Simplified 98.9%

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

   if -5.5 < z < 1.30000000000000005e-6

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 99.0

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

   5. Simplified 99.0%

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

Alternative 10: 61.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-55}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -5.2e-75) x (if (<= x 8.8e-55) (* y 0.0692910599291889) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.2e-75) {
		tmp = x;
	} else if (x <= 8.8e-55) {
		tmp = y * 0.0692910599291889;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-5.2d-75)) then
        tmp = x
    else if (x <= 8.8d-55) then
        tmp = y * 0.0692910599291889d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -5.2e-75) {
		tmp = x;
	} else if (x <= 8.8e-55) {
		tmp = y * 0.0692910599291889;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -5.2e-75:
		tmp = x
	elif x <= 8.8e-55:
		tmp = y * 0.0692910599291889
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -5.2e-75)
		tmp = x;
	elseif (x <= 8.8e-55)
		tmp = Float64(y * 0.0692910599291889);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -5.2e-75)
		tmp = x;
	elseif (x <= 8.8e-55)
		tmp = y * 0.0692910599291889;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -5.2e-75], x, If[LessEqual[x, 8.8e-55], N[(y * 0.0692910599291889), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -5.2e-75 or 8.7999999999999998e-55 < x

   1. Initial program 68.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 76.0%

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

   if -5.2e-75 < x < 8.7999999999999998e-55

   1. Initial program 75.2%

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

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

      3. accelerator-lowering-fma.f64 67.4

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

   5. Simplified 67.4%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 53.6

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

   8. Simplified 53.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-75}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{-55}:\\ \;\;\;\;y \cdot 0.0692910599291889\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.0% accurate, 47.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 70.7%

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

3. Taylor expanded in x around inf

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

5. Simplified 54.4%

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

Developer Target 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024196 
(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))))