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

Percentage Accurate: 68.2% → 99.5%

Time: 4.3s

Alternatives: 18

Speedup: 2.2×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 68.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.5%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-/.f64 99.4

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 99.4

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

   5. Applied rewrites 99.4%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(\left(\frac{6.012459259764103}{z} + 1\right) \cdot z, z, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       x
       (/
        (*
         y
         (+
          (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
          0.279195317918525))
        (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
      4e+296)
   (fma
    (/
     (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
     (fma (* (+ (/ 6.012459259764103 z) 1.0) z) z 3.350343815022304))
    y
    x)
   (fma 0.0692910599291889 y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 4e+296) {
		tmp = fma((fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma((((6.012459259764103 / z) + 1.0) * z), z, 3.350343815022304)), y, x);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))) <= 4e+296)
		tmp = fma(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma(Float64(Float64(Float64(6.012459259764103 / z) + 1.0) * z), z, 3.350343815022304)), y, x);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+296], N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] / N[(N[(N[(N[(6.012459259764103 / z), $MachinePrecision] + 1.0), $MachinePrecision] * z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))) < 3.99999999999999993e296

   1. Initial program 95.5%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if 3.99999999999999993e296 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))))

   1. Initial program 4.2%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.5

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

   4. Applied rewrites 98.5%

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

Alternative 3: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304} \leq 4 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.0692910599291889, z, 0.4917317610505968\right), z, 0.279195317918525\right)}{\mathsf{fma}\left(6.012459259764103 + z, z, 3.350343815022304\right)}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<=
      (+
       x
       (/
        (*
         y
         (+
          (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
          0.279195317918525))
        (+ (* (+ z 6.012459259764103) z) 3.350343815022304)))
      4e+296)
   (fma
    (/
     (fma (fma 0.0692910599291889 z 0.4917317610505968) z 0.279195317918525)
     (fma (+ 6.012459259764103 z) z 3.350343815022304))
    y
    x)
   (fma 0.0692910599291889 y x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x + ((y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304))) <= 4e+296) {
		tmp = fma((fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma((6.012459259764103 + z), z, 3.350343815022304)), y, x);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x + Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))) <= 4e+296)
		tmp = fma(Float64(fma(fma(0.0692910599291889, z, 0.4917317610505968), z, 0.279195317918525) / fma(Float64(6.012459259764103 + z), z, 3.350343815022304)), y, x);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x + N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+296], N[(N[(N[(N[(0.0692910599291889 * z + 0.4917317610505968), $MachinePrecision] * z + 0.279195317918525), $MachinePrecision] / N[(N[(6.012459259764103 + z), $MachinePrecision] * z + 3.350343815022304), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))) < 3.99999999999999993e296

   1. Initial program 95.5%

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

   3. Applied rewrites 99.7%

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

   if 3.99999999999999993e296 < (+.f64 x (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))))

   1. Initial program 4.2%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.5

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

   4. Applied rewrites 98.5%

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

Alternative 4: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(\left(-\frac{\frac{0.4046220386999212}{z} - 0.07512208616047561}{z}\right) + 0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.0005951669793454025, z, 0.0007936505811533442\right) \cdot z - 0.00277777777751721, z, 0.08333333333333323\right) \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma
    (+
     (- (/ (- (/ 0.4046220386999212 z) 0.07512208616047561) z))
     0.0692910599291889)
    y
    x)
   (if (<= z 2.5)
     (+
      (*
       (fma
        (-
         (* (fma -0.0005951669793454025 z 0.0007936505811533442) z)
         0.00277777777751721)
        z
        0.08333333333333323)
       y)
      x)
     (+ x (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma((-(((0.4046220386999212 / z) - 0.07512208616047561) / z) + 0.0692910599291889), y, x);
	} else if (z <= 2.5) {
		tmp = (fma(((fma(-0.0005951669793454025, z, 0.0007936505811533442) * z) - 0.00277777777751721), z, 0.08333333333333323) * y) + x;
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(Float64(Float64(-Float64(Float64(Float64(0.4046220386999212 / z) - 0.07512208616047561) / z)) + 0.0692910599291889), y, x);
	elseif (z <= 2.5)
		tmp = Float64(Float64(fma(Float64(Float64(fma(-0.0005951669793454025, z, 0.0007936505811533442) * z) - 0.00277777777751721), z, 0.08333333333333323) * y) + x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(N[((-N[(N[(N[(0.4046220386999212 / z), $MachinePrecision] - 0.07512208616047561), $MachinePrecision] / z), $MachinePrecision]) + 0.0692910599291889), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 2.5], N[(N[(N[(N[(N[(N[(-0.0005951669793454025 * z + 0.0007936505811533442), $MachinePrecision] * z), $MachinePrecision] - 0.00277777777751721), $MachinePrecision] * z + 0.08333333333333323), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 36.9%

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

   3. Applied rewrites 49.3%

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

   4. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if -5.4000000000000004 < z < 2.5

   1. Initial program 99.7%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{692910599291889}{10000000000000000}}, y, x\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 59.5%

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

   7. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 99.5

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

   9. Applied rewrites 99.5%

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

       1. lift-fma.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 99.5

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

   11. Applied rewrites 99.5%

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

   if 2.5 < z

   1. Initial program 39.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(\left(-\frac{\frac{0.4046220386999212}{z} - 0.07512208616047561}{z}\right) + 0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.0005951669793454025, z, 0.0007936505811533442\right) \cdot z - 0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma
    (+
     (- (/ (- (/ 0.4046220386999212 z) 0.07512208616047561) z))
     0.0692910599291889)
    y
    x)
   (if (<= z 2.5)
     (fma
      (fma
       (-
        (* (fma -0.0005951669793454025 z 0.0007936505811533442) z)
        0.00277777777751721)
       z
       0.08333333333333323)
      y
      x)
     (+ x (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma((-(((0.4046220386999212 / z) - 0.07512208616047561) / z) + 0.0692910599291889), y, x);
	} else if (z <= 2.5) {
		tmp = fma(fma(((fma(-0.0005951669793454025, z, 0.0007936505811533442) * z) - 0.00277777777751721), z, 0.08333333333333323), y, x);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(Float64(Float64(-Float64(Float64(Float64(0.4046220386999212 / z) - 0.07512208616047561) / z)) + 0.0692910599291889), y, x);
	elseif (z <= 2.5)
		tmp = fma(fma(Float64(Float64(fma(-0.0005951669793454025, z, 0.0007936505811533442) * z) - 0.00277777777751721), z, 0.08333333333333323), y, x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(N[((-N[(N[(N[(0.4046220386999212 / z), $MachinePrecision] - 0.07512208616047561), $MachinePrecision] / z), $MachinePrecision]) + 0.0692910599291889), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 2.5], N[(N[(N[(N[(N[(-0.0005951669793454025 * z + 0.0007936505811533442), $MachinePrecision] * z), $MachinePrecision] - 0.00277777777751721), $MachinePrecision] * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 36.9%

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

   3. Applied rewrites 49.3%

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

   4. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if -5.4000000000000004 < z < 2.5

   1. Initial program 99.7%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if 2.5 < z

   1. Initial program 39.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 6: 99.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(\left(-\frac{\frac{0.4046220386999212}{z} - 0.07512208616047561}{z}\right) + 0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0007936505811533442 \cdot z - 0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma
    (+
     (- (/ (- (/ 0.4046220386999212 z) 0.07512208616047561) z))
     0.0692910599291889)
    y
    x)
   (if (<= z 2.5)
     (fma
      (fma
       (- (* 0.0007936505811533442 z) 0.00277777777751721)
       z
       0.08333333333333323)
      y
      x)
     (+ x (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma((-(((0.4046220386999212 / z) - 0.07512208616047561) / z) + 0.0692910599291889), y, x);
	} else if (z <= 2.5) {
		tmp = fma(fma(((0.0007936505811533442 * z) - 0.00277777777751721), z, 0.08333333333333323), y, x);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(Float64(Float64(-Float64(Float64(Float64(0.4046220386999212 / z) - 0.07512208616047561) / z)) + 0.0692910599291889), y, x);
	elseif (z <= 2.5)
		tmp = fma(fma(Float64(Float64(0.0007936505811533442 * z) - 0.00277777777751721), z, 0.08333333333333323), y, x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(N[((-N[(N[(N[(0.4046220386999212 / z), $MachinePrecision] - 0.07512208616047561), $MachinePrecision] / z), $MachinePrecision]) + 0.0692910599291889), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 2.5], N[(N[(N[(N[(0.0007936505811533442 * z), $MachinePrecision] - 0.00277777777751721), $MachinePrecision] * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 36.9%

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

   3. Applied rewrites 49.3%

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

   4. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if -5.4000000000000004 < z < 2.5

   1. Initial program 99.7%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if 2.5 < z

   1. Initial program 39.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 7: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, 0.0692910599291889 \cdot y\right)\\ \mathbf{elif}\;z \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0007936505811533442 \cdot z - 0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (+ x (fma (/ y z) 0.07512208616047561 (* 0.0692910599291889 y)))
   (if (<= z 2.5)
     (fma
      (fma
       (- (* 0.0007936505811533442 z) 0.00277777777751721)
       z
       0.08333333333333323)
      y
      x)
     (+ x (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = x + fma((y / z), 0.07512208616047561, (0.0692910599291889 * y));
	} else if (z <= 2.5) {
		tmp = fma(fma(((0.0007936505811533442 * z) - 0.00277777777751721), z, 0.08333333333333323), y, x);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = Float64(x + fma(Float64(y / z), 0.07512208616047561, Float64(0.0692910599291889 * y)));
	elseif (z <= 2.5)
		tmp = fma(fma(Float64(Float64(0.0007936505811533442 * z) - 0.00277777777751721), z, 0.08333333333333323), y, x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(x + N[(N[(y / z), $MachinePrecision] * 0.07512208616047561 + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5], N[(N[(N[(N[(0.0007936505811533442 * z), $MachinePrecision] - 0.00277777777751721), $MachinePrecision] * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 36.9%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, \frac{y \cdot 0.07512208616047561}{z}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if -5.4000000000000004 < z < 2.5

   1. Initial program 99.7%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if 2.5 < z

   1. Initial program 39.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 8: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;x + \mathsf{fma}\left(\frac{y}{z}, 0.07512208616047561, 0.0692910599291889 \cdot y\right)\\ \mathbf{elif}\;z \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (+ x (fma (/ y z) 0.07512208616047561 (* 0.0692910599291889 y)))
   (if (<= z 2.5)
     (fma (fma -0.00277777777751721 z 0.08333333333333323) y x)
     (+ x (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = x + fma((y / z), 0.07512208616047561, (0.0692910599291889 * y));
	} else if (z <= 2.5) {
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = Float64(x + fma(Float64(y / z), 0.07512208616047561, Float64(0.0692910599291889 * y)));
	elseif (z <= 2.5)
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(x + N[(N[(y / z), $MachinePrecision] * 0.07512208616047561 + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5], N[(N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 36.9%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

      \[\leadsto x + \color{blue}{\mathsf{fma}\left(0.0692910599291889, y, \frac{y \cdot 0.07512208616047561}{z}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if -5.4000000000000004 < z < 2.5

   1. Initial program 99.7%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if 2.5 < z

   1. Initial program 39.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 9: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;x + \mathsf{fma}\left(0.0692910599291889, y, \frac{y \cdot 0.07512208616047561}{z}\right)\\ \mathbf{elif}\;z \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (+ x (fma 0.0692910599291889 y (/ (* y 0.07512208616047561) z)))
   (if (<= z 2.5)
     (fma (fma -0.00277777777751721 z 0.08333333333333323) y x)
     (+ x (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = x + fma(0.0692910599291889, y, ((y * 0.07512208616047561) / z));
	} else if (z <= 2.5) {
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = Float64(x + fma(0.0692910599291889, y, Float64(Float64(y * 0.07512208616047561) / z)));
	elseif (z <= 2.5)
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(x + N[(0.0692910599291889 * y + N[(N[(y * 0.07512208616047561), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.5], N[(N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 36.9%

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

   2. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. lower-fma.f64 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 99.2

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

   4. Applied rewrites 99.2%

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

   if -5.4000000000000004 < z < 2.5

   1. Initial program 99.7%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if 2.5 < z

   1. Initial program 39.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 10: 99.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.07512208616047561}{z} + 0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma (+ (/ 0.07512208616047561 z) 0.0692910599291889) y x)
   (if (<= z 2.5)
     (fma (fma -0.00277777777751721 z 0.08333333333333323) y x)
     (+ x (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(((0.07512208616047561 / z) + 0.0692910599291889), y, x);
	} else if (z <= 2.5) {
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(Float64(Float64(0.07512208616047561 / z) + 0.0692910599291889), y, x);
	elseif (z <= 2.5)
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(N[(N[(0.07512208616047561 / z), $MachinePrecision] + 0.0692910599291889), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[z, 2.5], N[(N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 36.9%

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

   3. Applied rewrites 49.3%

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

   4. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\frac{751220861604756070699018739433}{10000000000000000000000000000000}}{z} + \frac{692910599291889}{10000000000000000}, y, x\right) \]

      5. lower-/.f64 99.2

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

   6. Applied rewrites 99.2%

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

   if -5.4000000000000004 < z < 2.5

   1. Initial program 99.7%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if 2.5 < z

   1. Initial program 39.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 11: 98.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.00277777777751721, z, 0.08333333333333323\right), y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma 0.0692910599291889 y x)
   (if (<= z 2.5)
     (fma (fma -0.00277777777751721 z 0.08333333333333323) y x)
     (+ x (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 2.5) {
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 2.5)
		tmp = fma(fma(-0.00277777777751721, z, 0.08333333333333323), y, x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 2.5], N[(N[(-0.00277777777751721 * z + 0.08333333333333323), $MachinePrecision] * y + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 36.9%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if -5.4000000000000004 < z < 2.5

   1. Initial program 99.7%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if 2.5 < z

   1. Initial program 39.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 12: 98.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 2.5:\\ \;\;\;\;x + 0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma 0.0692910599291889 y x)
   (if (<= z 2.5)
     (+ x (* 0.08333333333333323 y))
     (+ x (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 2.5) {
		tmp = x + (0.08333333333333323 * y);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 2.5)
		tmp = Float64(x + Float64(0.08333333333333323 * y));
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 2.5], N[(x + N[(0.08333333333333323 * y), $MachinePrecision]), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 36.9%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if -5.4000000000000004 < z < 2.5

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

      1. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

   if 2.5 < z

   1. Initial program 39.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 13: 98.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + 0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma 0.0692910599291889 y x)
   (if (<= z 2.5)
     (fma 0.08333333333333323 y x)
     (+ x (* 0.0692910599291889 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 2.5) {
		tmp = fma(0.08333333333333323, y, x);
	} else {
		tmp = x + (0.0692910599291889 * y);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 2.5)
		tmp = fma(0.08333333333333323, y, x);
	else
		tmp = Float64(x + Float64(0.0692910599291889 * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 2.5], N[(0.08333333333333323 * y + x), $MachinePrecision], N[(x + N[(0.0692910599291889 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4000000000000004

   1. Initial program 36.9%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.7

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

   4. Applied rewrites 98.7%

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

   if -5.4000000000000004 < z < 2.5

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

   if 2.5 < z

   1. Initial program 39.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

Alternative 14: 98.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;z \leq 2.5:\\ \;\;\;\;\mathsf{fma}\left(0.08333333333333323, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -5.4)
   (fma 0.0692910599291889 y x)
   (if (<= z 2.5) (fma 0.08333333333333323 y x) (fma 0.0692910599291889 y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -5.4) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (z <= 2.5) {
		tmp = fma(0.08333333333333323, y, x);
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -5.4)
		tmp = fma(0.0692910599291889, y, x);
	elseif (z <= 2.5)
		tmp = fma(0.08333333333333323, y, x);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -5.4], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[z, 2.5], N[(0.08333333333333323 * y + x), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4000000000000004 or 2.5 < z

   1. Initial program 38.0%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.6

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

   4. Applied rewrites 98.6%

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

   if -5.4000000000000004 < z < 2.5

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

Alternative 15: 82.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+155}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0692910599291889, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
   (if (<= t_0 (- INFINITY))
     (fma 0.0692910599291889 y x)
     (if (<= t_0 -1e+155)
       (* 0.08333333333333323 y)
       (fma 0.0692910599291889 y x)))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = fma(0.0692910599291889, y, x);
	} else if (t_0 <= -1e+155) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = fma(0.0692910599291889, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = fma(0.0692910599291889, y, x);
	elseif (t_0 <= -1e+155)
		tmp = Float64(0.08333333333333323 * y);
	else
		tmp = fma(0.0692910599291889, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.0692910599291889 * y + x), $MachinePrecision], If[LessEqual[t$95$0, -1e+155], N[(0.08333333333333323 * y), $MachinePrecision], N[(0.0692910599291889 * y + x), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -inf.0 or -1.00000000000000001e155 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64)))

   1. Initial program 66.0%

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

   2. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-fma.f64 82.4

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

   4. Applied rewrites 82.4%

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

   if -inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -1.00000000000000001e155

   1. Initial program 99.4%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around -inf

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

   6. Applied rewrites 64.5%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 51.3%

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

   10. Taylor expanded in z around 0

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

       1. lift-*.f64 77.3

          \[\leadsto 0.08333333333333323 \cdot y \]

   12. Applied rewrites 77.3%

       \[\leadsto 0.08333333333333323 \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 65.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(\left(z \cdot 0.0692910599291889 + 0.4917317610505968\right) \cdot z + 0.279195317918525\right)}{\left(z + 6.012459259764103\right) \cdot z + 3.350343815022304}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{elif}\;t\_0 \leq -2 \cdot 10^{+80}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{elif}\;t\_0 \leq 1000000:\\ \;\;\;\;x\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+290}:\\ \;\;\;\;0.08333333333333323 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (*
           y
           (+
            (* (+ (* z 0.0692910599291889) 0.4917317610505968) z)
            0.279195317918525))
          (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))
   (if (<= t_0 (- INFINITY))
     (* 0.0692910599291889 y)
     (if (<= t_0 -2e+80)
       (* 0.08333333333333323 y)
       (if (<= t_0 1000000.0)
         x
         (if (<= t_0 4e+290)
           (* 0.08333333333333323 y)
           (* 0.0692910599291889 y)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = 0.0692910599291889 * y;
	} else if (t_0 <= -2e+80) {
		tmp = 0.08333333333333323 * y;
	} else if (t_0 <= 1000000.0) {
		tmp = x;
	} else if (t_0 <= 4e+290) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = 0.0692910599291889 * y;
	} else if (t_0 <= -2e+80) {
		tmp = 0.08333333333333323 * y;
	} else if (t_0 <= 1000000.0) {
		tmp = x;
	} else if (t_0 <= 4e+290) {
		tmp = 0.08333333333333323 * y;
	} else {
		tmp = 0.0692910599291889 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304)
	tmp = 0
	if t_0 <= -math.inf:
		tmp = 0.0692910599291889 * y
	elif t_0 <= -2e+80:
		tmp = 0.08333333333333323 * y
	elif t_0 <= 1000000.0:
		tmp = x
	elif t_0 <= 4e+290:
		tmp = 0.08333333333333323 * y
	else:
		tmp = 0.0692910599291889 * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(Float64(Float64(Float64(z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / Float64(Float64(Float64(z + 6.012459259764103) * z) + 3.350343815022304))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(0.0692910599291889 * y);
	elseif (t_0 <= -2e+80)
		tmp = Float64(0.08333333333333323 * y);
	elseif (t_0 <= 1000000.0)
		tmp = x;
	elseif (t_0 <= 4e+290)
		tmp = Float64(0.08333333333333323 * y);
	else
		tmp = Float64(0.0692910599291889 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * ((((z * 0.0692910599291889) + 0.4917317610505968) * z) + 0.279195317918525)) / (((z + 6.012459259764103) * z) + 3.350343815022304);
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = 0.0692910599291889 * y;
	elseif (t_0 <= -2e+80)
		tmp = 0.08333333333333323 * y;
	elseif (t_0 <= 1000000.0)
		tmp = x;
	elseif (t_0 <= 4e+290)
		tmp = 0.08333333333333323 * y;
	else
		tmp = 0.0692910599291889 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(N[(N[(N[(z * 0.0692910599291889), $MachinePrecision] + 0.4917317610505968), $MachinePrecision] * z), $MachinePrecision] + 0.279195317918525), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(z + 6.012459259764103), $MachinePrecision] * z), $MachinePrecision] + 3.350343815022304), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(0.0692910599291889 * y), $MachinePrecision], If[LessEqual[t$95$0, -2e+80], N[(0.08333333333333323 * y), $MachinePrecision], If[LessEqual[t$95$0, 1000000.0], x, If[LessEqual[t$95$0, 4e+290], N[(0.08333333333333323 * y), $MachinePrecision], N[(0.0692910599291889 * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -inf.0 or 4.00000000000000025e290 < (/.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 3.0%

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

   3. Applied rewrites 60.4%

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

   4. Taylor expanded in x around -inf

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

   6. Applied rewrites 14.2%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 1.7%

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

   10. Taylor expanded in z around inf

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

       1. lower-*.f64 57.3

          \[\leadsto 0.0692910599291889 \cdot y \]

   12. Applied rewrites 57.3%

       \[\leadsto 0.0692910599291889 \cdot y \]

   if -inf.0 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < -2e80 or 1e6 < (/.f64 (*.f64 y (+.f64 (*.f64 (+.f64 (*.f64 z #s(literal 692910599291889/10000000000000000 binary64)) #s(literal 307332350656623/625000000000000 binary64)) z) #s(literal 11167812716741/40000000000000 binary64))) (+.f64 (*.f64 (+.f64 z #s(literal 6012459259764103/1000000000000000 binary64)) z) #s(literal 104698244219447/31250000000000 binary64))) < 4.00000000000000025e290

   1. Initial program 99.4%

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in x around -inf

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 50.7%

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

   10. Taylor expanded in z around 0

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

       1. lift-*.f64 64.6

          \[\leadsto 0.08333333333333323 \cdot y \]

   12. Applied rewrites 64.6%

       \[\leadsto 0.08333333333333323 \cdot y \]

   if -2e80 < (/.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))) < 1e6

   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. Taylor expanded in x around inf

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

   4. Applied rewrites 71.2%

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

Alternative 17: 59.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-87}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+69}:\\ \;\;\;\;0.0692910599291889 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e-87) x (if (<= x 2.6e+69) (* 0.0692910599291889 y) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-87) {
		tmp = x;
	} else if (x <= 2.6e+69) {
		tmp = 0.0692910599291889 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.5d-87)) then
        tmp = x
    else if (x <= 2.6d+69) then
        tmp = 0.0692910599291889d0 * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-87) {
		tmp = x;
	} else if (x <= 2.6e+69) {
		tmp = 0.0692910599291889 * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5e-87:
		tmp = x
	elif x <= 2.6e+69:
		tmp = 0.0692910599291889 * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e-87)
		tmp = x;
	elseif (x <= 2.6e+69)
		tmp = Float64(0.0692910599291889 * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.5e-87)
		tmp = x;
	elseif (x <= 2.6e+69)
		tmp = 0.0692910599291889 * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e-87], x, If[LessEqual[x, 2.6e+69], N[(0.0692910599291889 * y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000008e-87 or 2.6000000000000002e69 < x

   1. Initial program 69.3%

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

   2. Taylor expanded in x around inf

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

   4. Applied rewrites 73.4%

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

   if -1.50000000000000008e-87 < x < 2.6000000000000002e69

   1. Initial program 67.1%

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

   3. Applied rewrites 80.2%

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

   4. Taylor expanded in x around -inf

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

   6. Applied rewrites 55.9%

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

   7. Taylor expanded in y around inf

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

   9. Applied rewrites 35.3%

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

   10. Taylor expanded in z around inf

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

       1. lower-*.f64 46.0

          \[\leadsto 0.0692910599291889 \cdot y \]

   12. Applied rewrites 46.0%

       \[\leadsto 0.0692910599291889 \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 50.2% accurate, 29.9× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 68.2%

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

2. Taylor expanded in x around inf

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

4. Applied rewrites 50.2%

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

Reproduce

herbie shell --seed 2025112 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, B"
  :precision binary64
  (+ x (/ (* y (+ (* (+ (* z 0.0692910599291889) 0.4917317610505968) z) 0.279195317918525)) (+ (* (+ z 6.012459259764103) z) 3.350343815022304))))